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THE UNIVERSITY 


OF ILLINOIS 


LIBRARY 


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FIELD ENGINEERING 


A HANDBOOK OF THE 


THEORY AND PRACTICE OF RAILWAY SURVEYING, 
LOCATION AND CONSTRUCTION 


BY 


WILLIAM H. SEARLES, C.E. 


Member American Society of Civil Engineers 


SEVENTEENTH EDITION, REVISED AND ENLARGED 
TOTAL ISSUE FIFTY-THREE THOUSAND _ 


BY 
WILLIAM H. SEARLES, C.E. 


AND 


HOWARD CHAPIN IVES, C.E. 


Professor of Railroad Engineering, in the Worcester 
_ Polytechnic Institute. 


VOLUME I 
TEXT 


NEW YORK 
JOHN WILEY & SONS, Ine. 
Lonpon: CHAPMAN & HALL, Limrtep 
1915 


Copyright, 1880 
BY . 
JOHN WILEY & SONS 


Copyright renewed, 1908 
By JOHN WILEY & SONS 


Copyright, 1915 
BY WiuuIAmM H. SEARLES 
and 
HOWARD CHAPIN IVES 


Composition and Electrotyping — 
By THE SCIENTIFIC PRESS (Robert Drummond & Co.), Brooklyn, N. Y. 
Printing and Binding 
By BRAUNWORTH & Co., Brooklyn, N. Y. 


PREFACE TO THE SEVENTEENTH 
HDITION 





Tue changes in this edition have been 80 extensive as to 
require a resetting of the entire text. These may be classified 
as follows: 1. Minor changes in the subject matter; 2. A 
rearrangement of certain. portions of the book; 3. The 
abridgment of certain sections, and 4. The addition of a 
large amount of new material. 

The net result of these changes has been to increase the 
book by. about 150 pages. The needs of both the engineer 


~ in the field and the student have been kept in mind in the 


work of revision. 
In Chapters I, II, IV, V, VI, X, XVIII and XIX while 


~. the changes made are few, they are nevertheless important. 
' There may be specially mentioned: the organization of the 


— 


engineer corps; a section on metric curves; work on paper 
‘location. The discussion of the Valvoid has been retained. 


— This subject, while somewhat difficult, admits of great possi- 


Ss 


bilities in the field. 


p jit _ Chapter III has been rewritten. The principles here given 


3) 
\ 


C\ 


have played and are playing an important part in the proper 
selection of grades and curves. 
Chapter VII, Reversed Curves, Chapter IX, The Spiral 


~ Curve, Chapter XIII, Earthwork Tables, Chapter XIV, 


Earthwork Diagrams,-and Chapter XV, Haul and the Mass 
Diagram, are nearly or entirely new. 

Chapter VIII, Turnouts and Crossings, has been rewritten 
and extended, as likewise Chapter XII, Calculation of Earth- 
work. 

The Chapter on Construction in the previous edition has 
been separated, and new material added, forming three 


- chapters—Chapter XI, Cross-sections, Chapter XVI, Con- 


struction, and Chapter XVII, Track Laying. 


3226890 


Iv PREFACE 


Acknowledgments are made to: The American Railway 
Engineering Association for permission to use certain material 
as noted in the text; to the late A. M. Wellington’s “ Railway » 
Earthwork,” which was the first thorough treatment of 
Earthwork Diagrams; and to Mr. A. 8. Crandon, for assist- 
ance on the illustrations and computations. Othef ac- 
knowledgments in connection with the Tables are made in 
Preface to Volume II. 

W. ss: 
Hy Celt 
Juty, 1915. 


PREFACE TO THE FIRST EDITION 


AurnoucH the modern railway system is but about fifty 
years old, yet its growth has been so rapid, and the progress 
in the science of railway construction so great, as to render 
the earlier technical books on this subject inadequate to 
the needs of the engineer of to-day. 

In the course of his practical experience as a railway 
engineer, the author was strongly impressed with the want 
of a more complete hand-book for field use, and finally con- 
cluded, at the solicitation of his friends, to undertake the 
preparation of the present volume. 

The aim in this work has been: 

First.—To present the general subject of railway field 
work in a progressive and logical order, for the benefit of 
beginners. 

Second.—To classify the various problems presented, so 
that they may be readily referred to. 

Third.—To embrace discussions of all the more important 
practical questions while avoiding matters non-essential. — 

Fourth—To employ throughout the work a uniform and 
systematic notation, easily understood and remembered, so 
that after one perusal the formulas may be intelligible at a 
glance wherever referred to. 

Fifth.—To express the resulting formula of every problem 
in the shape best adapted to convenient numerical compu- 
tation. 

Sixzith.—To furnish a large variety of useful tables, more 
complete and extended than any heretofore published, 
especially adapted to the wants of the field engineer. 

An elementary knowledge of algebra, geometry and trig- 
onometry on the part of the reader has been taken for granted, 
as a command of these instrumentalities is deemed essential 
to the education of the civil engineer. The few references 

Vv 


vl + PREFACE 


to mechanics, analytical geometry, optics and the calculus 
may be assumed correct by those not conversant with these 
branches. 

Many of the problems in curves are new, yet there is 
hardly one that has not presented itself to the author in the 
course of his practice. The investigation of the valvoid 
curve is original, and though the mathematical discussion 
is somewhat difficult, yet the resulting formulas taken in 
connection with Table VII, are exceedingly simple and con- 
venient for the solution of a certain class of problems. 

The treatment of compound curves is novel and exhaustive. 
A few general equations are established, which, by slight 
modifications, solve all the problems that can occur. 

No discussion of reversed curves is given, because these 
are inconsistent with good practice, except in turnouts, 
under which head they are noticed. 

The chapter on leveling includes a discussion of stadia 
measurements, with practical formulas. The chapter on 
earthwork contains a review of several methods for calculating 
quantities, and states the conditions under which these 
succeed or fail in giving correct results. 

Among the tables, numbers 2, 3, 7, 26, 29 and 43 are 
original. The adoption of versed sines and external secants 
throughout the work, wherever these would simplify the - 
formulas, rendered necessary the preparation of tables of 
these functions. The tables of logarithmic versed sines 
and external secants has been computed from ten-place 
logarithmic tables of sines and tangents, so that the last 
decimal is to be relied on, and no pains have been spared 
to make the table thoroughly accurate. 

Tables numbers 1, 4, 5, 6, 8, 9, 14, 22 and 30 have been 
recalculated, enlarged, and some of them carried to more 
decimal places than similar tables heretofore published. 
The intention has been to give one more decimal than usual, 
so that in any combination of figures the result of calcula- 
tion might be reliable to the last figure usually required. 

The tables which have been compiled and rearranged are 
numbers 15, 16, 17, 24, 25, 41, 42 and 44. The tables of 
log sines and tangents here given are the only six-place 
tables which give the differences correctly for seconds. 
The table of logarithms of numbers is accompanied by a 


PREFACE vii 


_ complete table of proportional parts, which greatly facilitates 
interpolation for the fifth and sixth figures. 

In all the tables, whether new or old, scrupulous care has 
been taken to make the last figure correct, and the greatest 
diligence has been exercised by various checks and compari- 
sons to eliminate every error. It is, therefore, hoped and 
believed that a very high degree of accuracy has been ob- 
tained, and that these tables will be found to stand second 
to none in this respect. 

The preparation of this work has extended over several 
years, as time could be spared to it from other engagements. 
It is, therefore, the expression of deliberate thought, based 
on experience, and as such is submitted to the judgment of 
brother engineers. If it shall prove to have even partially 
_ met the aim herein announced, and so shall serve to smooth 
the way of the ambitious student, or to assist the expert 
in his responsible duties, the labors of the author will not 
have been in vain. 

Wn. H. Searues, C.E. 


New York, March Ist, 1880. 


A kat 
yitde Ae 
oe ° 





CONTENTS 





CHAPTER I 


: RECONNAISSANCE 
SECTION = PAGE 
Ee EOPOSTADMICA! CONSIOCTAUIONS: occ. ik ncn ls abe een. cre pad 1 
Bee OLIN Doerr y cles a. ota Siete ake Weieer se aa a eae ee Cat Eee 3 
7. Pocket instruments......... ST Sie eae A Pe the Ae: 3 
SINE ASINOL GALS Reente A Penk a Cy Lee NS SIMEON. ince aig Set eReE Shc aaa Re geyee 4 
le Mornay tas Lore AMeLOUs cot. eins o sioline si) casa an tee seer eterno tas eer tee 4 
OME GOGK er CV.Clicaeusy aan nres ities. tele cudtagiters sce) e.sh6, Meret ia ny Rare ERE tf 
Pe EISiaA LTC ICOIMDABS..). ssi c sen che nce piace succes Picea eee att leis eon 8 
CHAPTER II 
PRELIMINARY SURVEY 

ES APC HTT TONS ar dkiy ay eC ee Iei le ale Stee te thle velevele oan Sremanats 9 
kee RAE Or CONDE. an bi mra a attic hes ao Oks wete cua » deeeeene 9 
20. Locating engineer............ Heal fa RR NAAR sind tho PET Cees 9 
mee arrive ch Fee Med ek Viet PR va eet gS fac Shetek 10 
PRC MIALIINCT. ee Mine tee.) Rae Mee ee nce ape dG accnelgie.« Gide ener ceack 11 
ARR ACTON Cores Maceo Act eS eh be dn ty ocean iiss, bossa 6b, e RUnla miner ener eke 12 
Zoom HODOLTADNEr.. Acie fess 8 3 Paes ANON ane Eee ee 12 
pam OLe Teun a Re oat GO ao Wa RIN Saccktineige ote @ sys wathe «a oe tceeer sp coeieee 13 
UME FCOCMN ODE ae aad) Ra id Cute teh as mete cis ohotancivcn.cut eres Clare eetehtee 14 
ci, MAM aRep(etoheay oe Icich Oe Aveaten e se Ratras dy Nah GR ee cick ae eae eee ene OI ie She 15 
SPN WARN GTeN ah ay Page re Pl copie! Pare ee eee © At 0 be ear a, Sar 15 
ees bClOmleViGli nd: pase ett ari SMe A hath Gest elis A ask e-Spo ene alo eere ee 16 
Ore NO OC NE coer Waa ew eaten ence rere a RO ee Ale «ree toda o cece de suche.s News 16 
BASE TO MEUIMOIRGL OD Ls. earache nie, ite by es) Meld ase Pee a «+ ieee eg ote aes 17 
BOREL Reep LANG DOlORAd, asta waycict sat tlonat ober concuone iahes aelslic sec aleie fhe nere 17 
ei) eelteatranial Ge epeus hile been Hiner Seuces Siege it herr ot cu, ene Fi 
PE ee TALIS A PONE Se wumene Ade intestine ele Gi epaier a SiGe SE Sigcuoae ion abu sha kegei ede 18 
nud, CUNY IIE Soe Wiech Sizes oS, 2 Ok ae Re | Aa ren Rn on RES ns eae ee eee 18 
44. Obstacles to alinement and measurement................-4- 19 
AVES” EE NEU GR UTEP ee OR ee ARC ee CR ec CaCREC Tee ern APE 19 
Se HCH AL ea SMa LANG LO cna wee cers coe et fant ob Lowel epd hse Soo gues oh apsey Sielsaid 19 
ate Leer tO TOD LECTUS Ares, siete opel cee teat ce ire apd eckak cass iSite, © gth sin Reuters 20 
Pe GHeME LOTS ANGLO see ne Airy the cae Tacit wiccsce + a <3, ectiats & oe al 
OME SCLCOULOM TOL ATO LCR en. Sever cweie Ncnctore. xe ttusp cusindsas (ac 6.2866 10%6, 's cyoreee 
EME OCICVE SHOES \) LLO-UINCS ein. ade healt odd a Sh ona b Aleioh Aelcal «iiss ¥ eth o-96 eras 23 
54. System of plotting Map eee eels cae A REP gia Gee INSIST. Sta star's . 24 


bs CONTENTS 
CHAPTER III 
Turory OF Maximum Economy IN GRADES AND CURVES 
SECTION PAGE 
DOs GHOICEVOL TOUTES cece ince ok how cote Eee eee 26 
56, otatement of the prohlem. ie tin. 5 ogee ot cee ce ae eee 26 
oe wraction of a locomotive... 7. «cece nee eee ee ee ae 27 
OSs) HN LINO iOXDCNSEL Wee. Moms, okeraps eee ee Ee ee 27 
HOw Resistances tO<moOtion yiah... uae soe eee an oe ee 28 
605 Resistances oniarstraroeht. Jevelstraclkoaesasn iene nieee cae ae 28 
615 Resistance: due to gradect: fests. terse cle ores oie en oe ee ee 29 
62a Resistance Cueto Curvy fo.s castes Shc. eore ee ie eet ReeT 30 
G2 Cury OxCOMPensalLion™ a. cmalmicessgels sts oe Reo ay Se aaa eee eee 30 
64 SVormiulassiordunaxilnU i Gralssuie ees clekel emcees ciel diem tere nena ete 31 
G5. sHngine-stage.. Fy atta apne wei as ek eee hosedaletore Cae OIC eceen cnc enT ae 33 
G60 Graphical: Solmbrow,.c aes tic ess) sie lotto ee eater alate a aie fae ier ce eee 34 
Siem brain. Oad reel procadls\., scusiae stee aietc s Giels shetatchete acne ace eee cee 35 
HSwmeeducnlon Ol @radesvoneCUrVvesscowerd< ene eesheteieiene icicle ie ace EP a 35 
BOR XATIPLEs J: 70 vide, 5. « Meaac oe 5 ik cae ee ao gue oe aR ees ue ee 36 
COwP UShersarad O82 cccic, A cepacia dal ae oteeet ae tareishelers wir stenseetencen ors eae 37 
flee NEAXIIM WINS LOGUEM er AG ese ae cr ea cree asec ete ae ay) 
To wound ulating gradesser nec a. aoe inte ieee eee 38 
VowCOMparison of TOUTES: PAs oe okt a eieiaialecrctensteher enone eter 39 
7670 V alue°of distance savedic...5-.00 s.¢ oe ii Glace oe 39 
Mifae @ ONCLUSIONS armen ets PER eA Re RNA ALOR HOG Cn) COIOIDOICIG a) ake, 
CHAPTER IV 
LocaTION 
fo CAXCNEFAL TETAATKS.. accrae «vis eale @ apa < Oe sree Wee ne brs diareueia le Senta e 
COM WONG LAM CeNtSs.) ace ais seston a cantons aes Fore saia le eusketer gnats Gree te rsictie ange Sil 
SOs Levelers duties Proimlingyecpecatesicieeie a ctererereheterenerele aiete ate 42 
81. Establishing grade lines...,...... hon Conoco noRbonk ek 
CHAPTER V 
: SIMPLE CURVES 
A. Elementary Relations 
89 Limits £0 curves and. tangents s. tes. 50-6 cletee wiser eeenene 44 
Rom Meinitvon Of termsgss si. ocaeteee a © cher © eokee eis acd ee eee 44 
S4 Radius and degree of Curve sac o. cs che ae cme ole bce ptetene meee 45 
Sh Weasurement of curvest. <. ...c's a ceeid ee eee eee en eee 46 
So Approximate value of fos 5.. Sees tincche Sistwiace Cusexcean ale eanennae eens 46 
s7. Central angle and length of curves) «+35. sks fe ee eee 47 
SS Dehnition of other elements: 2.5 2 7.% tom st cere neta a ete eee 
90, formula for*tangéent distance’ 720.4 >. 2. ohh cee eee ee ae 49 
Glo hormula for long ChordiC. na sta aces 2 ose ere 49 
09 sformuls tor middle ordinate Mor. : 5.0. 1 eieerettret eetennee 50 
03, Formula for external distance 2.337:0 ss were eee es bake eer COS 


CONTENTS 


SECTION 
95. Formula for radius in terms of T and A.......,.......05--. 
96. Formula for external distance in terms of JT and A.......... 
ay. Formule for radius initearms. of Band sA, < o icfoce dnc athens 
98. Formula for tangent distance in terms of EF and A........... 
pp. "To define. the curve of an old tragkio en ices Fh We wpep iach oth 
400, Other curve formulas; Table XLT. scectis 5 pheacrey ona doo owt 


FHi:: Deflespon' angles sh 4 Cte) 70. 00s. Te fen seem be ot 
102., Rule.for-deflectione aay im. saliwowlt mew 0), oxvive.e bak of 
103. Rule for finding direction of tangent at any point........... 
1044 Second imethodsot deflections: sa. uno at eee De ela 
105; Subchordsi sates ea 2S ORS OS hn, Oe eat ak 
106;_ Defiections*toreSubchords4ie Sree naa GAT OS. 
107- (Corréction=forsubehords) sate ets Oe ti hernias See 
108. Ratio of correction to excess of arC............. cece cece es 
HOO SE Ial ls DOCCS Peet Pera pet cet dae shed adanthar, Dap 
110. Method by: deflectionSkonl yt ageen nen | eeciber. ead. bad act 
BS NE CUTS GUIV CR TE teenies Gece hate ars wey ta ranazertns Ae SiN NES SE, eer 


C. Location of Curves by Offsets 


Lot Four methods. Peek Ta, Seek ee eae, TRL eR sd: 
118. By offsets from the chords produced.................0.00.. 
114° Do. beginning! with a-subehord ys. 72.4 “Purges ssc avaes.. 
J15. Formula for subchord offsets, approximate................. 
Bia Byamiddle ordinates... 2 2eb & Bra ap tecneee o a cis. sroty 6 ote cetera ome 
TUs= Do. beginning with a *sumechard ) wm eit vale se cocci os ener ste ewe 
MiG eB cangeent: OLSELR naa. ake meee crete ae eee. ciske Or oe ene lee 
1207 Den berinning with avaubehordiwsidtaed .-. 02 se ete ete oe store 
(ot Byoordinates froma lone Chord, ecre tn che telene cen «cere sts <a ComeeraNe 
122.4 Do. for an) even mM UimMbpenr Ol StattONst veg. hhi wd. os. ce ee alles os 
EZsme Most Oran Odd NUMDEr OL StatvlONS ss t.ae cm cis aie ics Hiaweie.ssa eer et more 
124. Do. for an even number of half stations................... 
125: Do-beginning*with any subchord: . . fs. 6 Peis eae eG 
126. Erecting perpendiculars without instrument................ 


D. Obstacles to the Location of Curves 


12 (ene: VertexaN accessible, v2", Aix Goce pk nore oh oe a a healed! hohe one 
1282 ¢itherpoint ofcuryv eunaccessibles 2 ete mae oe yk sk si aqetelsl sate Pecarecs 
129. The vertex and point of curve inaccessible. ...:........,.06 
130. The point of tangent inaccessible... . 0.0... 08. 0e eens ened 
131... Lo pass an ‘obstaclefomiaicurves dise fo Cth wed 


E. Special Problems in Simple Curves 


132. To find the change in R and E for a given change in T,.,... 
133. To find the change in R& and T for a given change in #,..... 


X11 CONTENTS 
SECTION ; PAGE 
134. To find the change in 7 and £ for a given change in R...... 80 
135. General expression for elementary ratios................... 80 
136. To find a new point of curve for a parallel tangent.......... 81 
137. To find a new radius for a parallel tangent................. 81 
138. To find new P.C. and new radius for a parallel tangent...... 82 
139. To find new tangent points for two parallel tangents........ 83 
140. To find new # and P.C. for new tangent at same P.T....... 85 
141. To find new P.C. for a new tangent from same vertex....... 86 
142. To find new radius for a new tangent from same vertex...... 86 
143. To find new R and P.C. for same external distance, but new A 86 
144. To find a curve to pass through a given point.............. 87 
145. To find new radius for a given radial offset................. 88 
146... Hquation.of. the valvold: i. 62 seme ele Sean aes wa 90 
147. To find direction of a tangent to the valvoid at any point.... 91 
148. To find the radius of curvature of the valvoid at any point... 91 
149. To find the length of arc of the valvoid................0.... 92 
150. To find new position of any stake for a new radius from 

SO TCD Pops Cra vo sols Papeete delet GRAM Coa ROO IE bene 93 
151. To find new radius from same P.C. for new position of any 

SEATION 2 cris le lesa pentusnchaae so eeyens oe eden ee a ee eee 95 
152. To find distance on any line between tangent and curve..... 97 
153. To find a tangent to pass through a distant point........... 98 
1540. o find a Jjine tangent to. two Curves... .0, cuits aun vole 100 
155. To find a line tangent two to curves reversed............... 102 
156. Study of location on preliminary map; Templets; Table of 

convenient. curves;, Paper locations: 1 sidweee seni ate 104 

CHAPTER VI 
CoMPOUND CURVES 
A. Theory of Compound Curves 

TS7 Ss -Definition.c sewn os eee bee a eal ie Sai ee ee sae ae 107 
158: Che,circumseribing circles, siete debe oe ee ae 107 
159. The locus of the point of compound curve.................. 108 
160. The inscribed circle of the principal radil.................. 109 

Cor. 2. - Maxima and minima of the radii...........,..... 109 

B. General Equations 

161. Formulas for radii, central angles, and sides................ 110 
162. Given:: Si, Se,-A, and Ri, to ind At, As, and Ret). bi doh... 1. 112 
163. Given: AB, VAB, VBA and Ro, to find Asx, Ai and Ri....... Bla 4 
164. Given: Ri, Ai, Re, Ao, to find the triangle VAB............. 112 
165. Given: A, the radii, and one side to find the other.......... ls: 
166. Given: one side, radius and central angle to find the others. . 114 
167. Remarks ‘on special!casés ).N)4 0. 0s. 0. Oo Ses a ee 114 
168. Obstacles; the P.C.C. inaccessible... 0.2.0.0... cece cece ee 115. 


CONTENTS X11 


C. Special Problems in Compound Curves 


SECTION PAGE 
160, To finds new 2.C:C. for a parallel tangents. a... ceeenue sere 116 
170. To find a new P.C.C. and last radius for a parallel tangent... 118 
171. To find a new P.C.C. and last radius for the same tangent... 121 
172. To find a new P.C.C. and last radius Re’ for new direction 
Of tangent brTOUCOESAMC Eyal pra che elation a's eh noes eee re 123 
173. To find a new P.C.C. and last radius Ri’ for new direction 
Ol saneent LHroug busses lege we eels aah ue late. eens 126 
174. To replace a simple curve by a three-centered compound curve 
between the same tangent points............-0.++e+eee- 128 
175. To find the distance between the middle points of a simple 
curve and three-cen tered COMMPOURCACUT VC) Sets i) 2) wen ae 130 
176. To replace a simple curve by a three-centered compound 
curve passing through the same middle point..... Pe 131 
I. The curve flattened at the tangents. ...............-6. 13t 
II. The curve sharpened at the tangents.................. 133 
177. To replace tangent by a curve compounded with the ad- 
ACEI Gs CUT VCS a carey eden s cadets tp iean cae cose co a ea 135 
I. When the perpendicular offset p is assumed............ 136 
II. When the angle @ or @ is assumed.................0-6- 137 
Li taawihen: the radius, its 18) aSSuUmMeds, 5 2. <hid tonsa as cals 138 
LVewLocussol the: center Oorrcraus gan. tet Bl a ee 139 
178. To replace the middle arc of a three-centered compound by 
ATE OlsGiLeren bt: TAGlUSe acre eed bees eect toe eit Aone 140 
I. When the radius of the middle arc is the greatest....... 140 
II. When the radius of the middle arc is the least.......... 140 
III, When the radius of the middle arc is intermediate,,,,,, 141 
CHAP T Dea Le 
REVERSED CURVES 
ZO. se hinihyONdes eee os ocd SF eee SUE BANS TI, ISS 146 
180: (Given p andr, to find *centralangle? oo 45 Ole, Aaa alee. 146 
Sd Givensen e771. ALG Az tO: Foret ee ate, NO A eee seenemeE Ne oe 147 
182 -Givenn 1), G7 ANG: fly COMIN T22 soe eee tee eee ede oh iseedid saevah 147 
A83-"Givensol Ar andrAs, to: find rn oe eae AA PO a 148 
184. Given: J, Ar and Ag, to find r, curve beginning and ending at 
GIVEN POUNLESS oa os Sec, Pe eee LD. RT UM, RMSE AES, 5 149 
185. Remarks on solution of Curve problemssiitk DLV Lanes. 150 
186. Given: A, distance P.C. to V, m and re, to find Ai, Az, and 
the- distance tromeVsto the: Pvc ae. SA ice 150 
187, Given: A, m1, v2, and the distance from P.T. to V, to find 
Ai, de, and the distance from the P,C, to Vivrererrerrre, LOL 
CHAPTER VIII 
TURNOUTS AND CROSSINGS 
ES SME SL EEIONS (OW St CHOStmrta acto, keroacet ct oie, cre seet cere, s. srshereleile) <'ai,chaus 153 
Exe at roge ety pos O15 TAM OOr OF ois ds aaa els «Seale e sieve. @ He 6.0 tas 154 
VOC were aalser ne UU Ol! Grae rere leva vc 5 © vase a biol'e «16.50 Ke sl ote suse 156 


X1V CONTENTS 
SECTION PAGE 
191. Split switch; radius and lead; lengths of closure rails........ 156 
192.5 Tables 1Or tUuYnOUUSeels ves occ er eee ee eee eer eee ee ain 157 
LOS LOA PERCEICAl LEAUS cts crac tte. Lait meetin aire ac ema a eee 158, 159 
LO5. ePendea SpllucswitGhiaccs. cues esnctco emer morte mes ence oe Een 160 
1906; Stub switch. raciue and 1oncs. sont ens ae are ee te ee eee 161 
197. Location of rails between switch and frog points............ 163 
LOSe, Track DEyYONd themeel Ol TrOm a... emi cient ae seem ee nee 165 
199, 200. Crossovers between parallel straight tracks........ 165, 166 
2Ol Lurnouc tO afparollel side vrav ks ataaeinat eee mera ence nee 167 
B02 AUS. AO GEr ANG DOM y) CAC Ka creat, ene ee aeen ns ieee ae een 168 
204, 205. Turnouts from curved tracks; stub switch............. 170 
206. Split switch turnouts from curved tracks, ..2 0.20... 173 
207. To connect a curved main track with a parallel siding:...... 173 
208. Crossover between parallel curved tracks.................. 175 
DOO. ‘Crossing or straight, anu curved” track... 2-2. see at cee 176 
210. Crossing of two curved tracks........ ee kr eine aes Oy Tees 177 
. 211. Turnout connecting two straight tracks which cross......... 178 
212. Crossover between non-parallel straight tracks.............. 179 
Ise Silt: SWALCH CLOSSIM cotter e5 acct abies ones eds Bee ee ee 180 
Dla Sy IMMetrical Y tlackSer a te.cco chitee tale cisterna ncieteree hereon means 182 
QoOwONON=sy MiInetHcalwYetrackaie cette ait at ciricnmne ne cmt niet oe 
CHAPTER IX 
Tur SpiraL Curvp 
DUGHAUibLuaL yaotsuhe Spilal CUT Viesen me uwertenacie J Gels aisreletcmnien eae temeaae 185 
Disk orm and design of thewspiralcurves ...-crs 5 + cient ar 186 
DIS Spiralsdevections and cOGrdinatessauiee: oni acta t nena eee 187 
219. Adaptation of the spiraban ‘praciiGe@sartl <2 ..e ese + esse eect 188 
PPO. Spiral long chords and, tangenigwer ncn. Seem eerie ane 188 
2p. Coordinates,p and g of thé, polit, Aaa halt al os ei a ee ~189 
Dev. Tangent distance for-curve with Spitalsy. 6 Sere «a1. 0 syne 190 
Ope. External, distance for curve: witlesplEplssn .ja.ter ws ¥- ye Pas 191 
DOA wali fference Of JesIN) Germs: Ol ie: ae Aare anes ek Shion sachs he eae ee 192 
225.282 ovapply ispiralsjto; asimple curve (45) sk em od tom otgse ee 192 
226) To.apply. spirals to. a.simplescurve (Ai=0).. - . . perce iw curnts eae + © 194 
B27. lo compound a-simplegeuky.edtormspinals, oy-4 3421 on. ws pabues was Re 195 
928.!To apply spirals to a simple curve (h =). csscs.b.sasenceesr 196 
a.2 Lhetspirals alik es piu, Se We ard oad Ts siccvnd’ arousye haces ke 196 
baa he spirals diferent qvmiteros nid tetas con csies cA) eee 197 
229. .To.apply spirals to a compound curve (h—)...............- 198 
230. To apply spirals to a compound curve (h+)................ 199 
231. To insert a spiral between arcs of a compound curve........ 202 
232, 


Hieldsandiofhice, works «siti: eet sues tue steers Fi wae US: 


CONTENTS XV 


CHAPTER X 
LEVELING 
SECTION PAGE 
Basse Olsthe engineer s/lOVvelier, a ca sheha' cretarateres osteo wns 6) oes «ore ohene 205 
Ol Ave Dire, datum eh OWsAssuIned a... <0 ccc ciekae cobs © Sie Rigas ties be eon 205 
Scie DENCHes me hOWalISCO sya tao Theme tneet sin ¢ aoe Mists c0 3 dela aisle ots 205 
DaKeetlelgmicominstri ments 11 of ae ere op peinic. ciem odueleia oaeloke ware ee ia vePe 206 
DAME CCO CIT P LOL t DEZTO Cae thts eae sth, hceetoteecOs bans iica ls eiiuer tn ae Gam 206 
238. Elevation of intermediate points...............005 wb ea ale 206 
Wee me Lerningy, POlmnts,. Vek wd cars SS Se kev ie 0 ENP ake RUNG EOE: 206 
240. Rules for backsights and foresights:.. 0. .iclo0ct.s.00050.05 0 207 
241. Form of field-book; proof of extemsions.................00. 207 
eae e TOMES a deea ects seas dene wen kin Sic Re, cee Ged gs bn fe 208 
240. simple leveling: Test. levels:in «seit taaloe fk sls ee 209 
244. Errors in reading, due to the level; how avoided............ 209 
245. Errors in reading, due to the rod; how avoided............. 209 
246 SH TrOrs. due. toscurvavure of the earthe.. ae . «nade ns so he sche 210 
BA GARE ETOLSLEAU GatONTeLEACCION Sev tic sivicerc chs a es cis eka weetie caecis 211 
CA Ren Ae scOlCUEVaALUTe Ol ENE Gartiltsnr sn wits. o iridieie a cketsss gis selene PAA 
VAG mE IL eVelin ab Vela Sl tat ye cor at ego iate as secur Sescee ecker sore oecietale 212 
250. To find the H.7. by observation of the horizon....:........ yi lias 
D51. Stadia meastréments: horizontal sights.::.................- 215 
252, Stadia measurements; inclined sights, vertical rod.,.,.,,... 217 


CHAPTER XI 


CROSS-SECTIONS 


53. Final profile; selection of points for cross-sections........... 219 
254, Rocenceading forsera dewey. Spans ees wt. ule Mie Mate « Setters 220 
255, Gross-secthions, LOrmulag for wee areas site oie tins wee ae snes 221 
256. Compoundictosssectipissstv.aiekic Were ADGA) so. 0 cscs cle ee oe 220 
MMICY Seat aleeCkIast ys citi actate ok Wns seu cus sinh oo bp bole. 224 
258. Other methods; use of level board..........sseeeeeees swe ees 
oe Hoplodevels sc 9 ae Re ool B08 ie nth pe osc Lae 225 
BGO OFM OL CrOss-SeCtlOnuDOOK are nie «taliadds 5.1/9 Fic ie csleknd ett meeo 

CHAPTER XII 
CALCULATION OF HARTHWORK 
261 a Prigmords: Choice, Of CrOSS-SECtION: «. 2 «2 sis « + +4 40's sfeue o.c)0's 227 
Dore OrmeUlas OP Sectlon al ALCASUA.meo ckasshers scsteteae Aeteetee eae crpreloli ns 228 
mos Orme lasitor solids COMLENtsc ka). ca tisthos, Sie ke! choncre aL Misnond ws n0fSys 232 
264. Prismoidal correction formulas.............cseceeceeceucs 232 
265. Application of the prismoidal correction. ............:..s%. 235 
266. Correction for curvature in earthwork...............e+eee00% 236 
Ou ESOLALCGEIMASSEM: cfc cssheretertiecene rs tone iGpolene aie rare. cle ware Geers oladeinte Bee's 240 
ESE OOS DOLTOW HDIGE ... 8 mar iatee oh enor nd scans ott Pie RASS: 240 
270. Volume of truncated triangular prisms...............e000- 241 


Xvi CONTENTS 


SECTION PAGE 
271. Volume of truncated rectangular prismsS...........-...e-06 242 
O72 Method iby: Unttjareds) oc. pierce reise oeaeke stele eisl ote e buen ete oa reeee 243 
BTSASOLD CT: NCGHOUS ethno sc ale clothe, ooauetete tel eeezenel ohovoiakanellore Gah del tenement 244 


CHAPTER XIII 


EARTHWORK TABLES 


214, Kinds, of tables.) 2s 27 ire cs > oSaitela Geeks te Ae Eee ee eee 245 
2475. Level=section ‘tables: 2 o>. sx scan ok eiees ee De erie cratic oe 245 
26. Lables of triangular prismssiie one i ate eae eet ee ee 246 
277. Tables of three-level sections; extension of level-section table 246 
278." lables: tor irregular SectlOnSsns.. 6: sendes2 ene selene eee 247 
29. -Prismoidal correction tables. ... saterels feat doe) taahdaa tenets eet Bnd ee 247 
280, .ixample, .: Wek iced | eer eal obits, Sees eet eee 248 


CHAPTER XIV 


EARTHWORK DIAGRAMS 


28 General prince plestoderts cis )a ccs ccteai casa ace Re econ eee 251 
282. Triangular PrisMIS5. 9 socks co ae eee Wo Le ne Eee 252, 
250 Eriangular prisine: use ol the ciaerann.. oe elie ena 203 
284. Diagram for three-level sections. o........¢.foscde- ccs eecce 253 
285. Use of three-level section diagram... .2 <.. sss cient eu cere 256 
286. Diagram for prismoidal *correctionsse. «se... >see selelens -. 206 
287. Use of prismoidal correction diagram...............se-eees 257 


28 Sa CONCIUSLOMN cor eis ccheloeesiene seh aes eo) rote UE ae Tet2bs 


CHAPTER XV 


Haut AND THE Mass D1aGRAM 


DSO P SEPALS ees Se Ne eek eae eel SRO etn en nc ees Aree eee TO eee OS MEET 259 
290. "QUANtItLY Profiler aise Sines ene ae ie Oe ate ee ee 261 
QOVord he Mass Caer aN fo se eae ee ee edhe ac eee eee 261 
Zoe. Pronvaple Haas 207% ie Wee = os eter he se sta one cee eee 264 
293>Payment Of oVerbatle evs. sate ete tees nc oe en eo nee 265 
294s EXAMPLE oo Peete ls COT ee He Miia atetodera fe aue sae ack nee > 266 


CHAPTER XVI 


CONSTRUCTION 
295. Organization of engineering department............... eee On 
ZOOM CLEATING ANG STUDOIN ET ose e eset ee co arene ene etcetera 267 
Zoi est evielsand 72Uard Plus aioe cso ee eee meter tete Tenet: 268 
ZIS SCTOSS-SCCUIONG 2 Wee ates oe eras ne een ier reine 268 
OO PATINA 45.5.5 SE Sade a > wine eae 0 Rt pickers ae Ps sane Oe 
300. Alteration of line..... TTI Was Foes ssidacnm ose 96% 88 270 


B01. *Drains and. culverta; - P00) Alte peas fo Bmploy 270 


315. 
316. 
317. 
318. 
319. 
320. 
321. 
322. 
323. 
324. 


325. 
326, 


327. 
328. 
329. 
330. 


331. 
332. 
033. 


CONTENTS XVli 


SECTION PAGE 
SOLMMATCHOCULVOTEGH. sek es ere emia a Chee ers Oe Oc ae ete a tea oR tne 272 
303. Foundation pits; Bridge chords on curves.................. 242 
4 ay Gate Sao URTC St rhe Ee teas beet Ge sues LOtEe crimes sue NCR a ete 277 
BOSmerestleswOrkn cera ctaetoce ts iene at toe: niet. ohare Sates 2770 
306. Tunnels: Location; Alinement; Shafts; Curves; Levels; 
Grades; Sections; Rate of Progress; Ventilation; Drain- 
DLO ee re ee eas. er aay Sela whee stat eae bras a che dsl abet Re 279 
SU CeeECCUrACIng: CHESING: okt sec. ee see eyolcrtie a seen at hes ol us 2 cL 285 
SSE STOCHCIECNES ANU PaAlNS oe tere Mitotic sy ahel shotchelye caont tered a orotate 286 
SO RaMOT IVA NOG ULINI A COnc ta oitcess arene eee ap se tare Meria talc = taste aes creat bathe Ries 286 
SL CMMIVEOTG MI VCS tLINACeS ect le cus cr sislel® acolo aren eesaaccea aie act tobe oaierans, 287 
CHAPTER XVII 
Track LAYING 
SE LACINOEDaLLASt aah re vee ca eae SAR on , eens eM) ee ree, 289 
Mees er (lencen MieretakeCs me waentd hits oq a pean outs oss Geetha crete aieane 289 
ais ae OMIT VIN ONE All Sy Ane toe, ere case ts Sea SES oie: itera gooey otetie age a nae 289 
314. To find the middle ordinate m, for one station in terms of D.. 289 


To find the middle ordinate m, for rails in terms of rail and R. 290 
Deserme trac kancurvalUrse. chart auiopel chor. Uae ttake avai eucsan: naira 292 
(EPS PCLAW CES esas een Pegr GUIS RNa Mc LENER A Ainge Sur RRR eee fo 293 
PVErtrCaloCUurmes nie wicreits . seer de ofetcda REL eee ce eee oe op Dene 293 
Dye beck gOS tae mb ome ery nde ers iy Oi Ahr s ince feet Re aes ae A A he 294 
Second: method tor.viertical Curvesiaela tae .cicesneen see 295 
eMiTdemet nod Ole Vertical CUrVCRe smi teas eee ok ee ema 296 
Wenethrote merical cuL ves rsa. Ulett ene chi ortnis seacorerens 297 
No find theelevation of outer rail on Curves)... .... 00sec ee. 297 
To find a chord whose middle ordinate equals the proper 

Chow a GO Msp seg bar os ects als vaktr- a tuokecak eM eU ae wie A Bee Basle ee 298 
General remarks on elevation of.rail.. ...%4..-...03 «004s 299 
Hp xpoerSLOMy Oler alls SsSlCID BS caumeiswueianed eal cgastary snare ers eee oO 

GCHAPRERE Ri ecV LEE 
TOPOGRAPHICAL SKETCHING 
(VET OT OL PROUT LCS an rcenn ett ateens cea tes cenclan asta te ete ilistiey sivas ote et Sucas 301 
INEST CITT LUCE Et Seca ete ree ca Meee tic cat cise tret er oke artic ee 301 
Natural features; Contours; Hatchings..............0.++-0. 302 
Methodlofgsketehin gst siriecs ln ve pete eels, cles Sr eet, Paeaihis BOOS 
CHAPTER XIX 
ADJUSTMENT OF INSTRUMENTS 

ST GEC LATISED eet ene st ee eee ael ant AL aeue Moles. wekeaoane tilele. a ene.s 304 
Flere, PAE cg SiR ae ea I Ue ee are Bs eae a Owen es 366 
The dumpy level....-::+eerees ere ee Sek Satie Sn eae 307 













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a : TE, ARK es: Ce y 3 
Pgee ge hE) (acre, 
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pe" 3 *, ae ene sted + “PSS en marie tire tints on ‘d ee as 


* 52) ee 


ie a ey ed Sa. ) { ah Cris ‘ os “dks 






















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Ay Salaam aa peas ite Sin 
ass hie cen Se Mave ah tas ia i: 
Be ESS Jy pes prints aitn Yak aes 
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pas th. vis habperltta sam aya’ Pisa A ANG a bd 
op as 5 {Ae y ee fr Tie are ins eng 
Ms Puy “FY et Aas pap tA S ie hee eer 
tig.) voles “hse Shas eae Pe , 
. i: a A 3 ygesdy st BAS 
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: rae homie he dledieen ose uribabe afta HORE 
Wee. init vied sie dewaieed savant Te 
Sais Fe hea Get heat pee ft beep ap “| anal ssfiee tpi 
Betae. RUS) Dear y ETS CS Tay oat eee 
Bes “i a: nad PE tae a Min RE =| . t 
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FIELD HNGINEERING 


CHAPTER I 


RECONNAISSANCE 


1. The engineering operations requisite to’ and preceding 
’ the construction of a railroad are in general: 

THE RECONNAISSANCE, 

THE PRELIMINARY SURVEY, and 

THE Location. 

2. The Reconnaissance is a general and somewhat hasty 
examination of the country through which the proposed road 
is to pass, for the purpose of noting its more prominent 
features, and acquiring a general knowledge of its topography 
with reference to the selection of a suitable route. The 
judicious selection of a route may be a very simple or com- 
plex problem, depending on the character of the topography, 
and more especially on the direction of the streams and ridges 
as compared with the general direction of the proposed road. 

3. A road running along a water-course is most easily 
located. In this case the choice is to be made merely between 
the two banks of the stream, or between keeping one bank 
continuously and making occasional crossings. When the 
stream is small it will usually be found best to eross it at 
intervals, the advantage of direct alinement outweighing the 
cost of bridging; but when the stream is of considerable size 
the solution of the problem is not so obvious, requiring patient 
comparison of results in the two cases to determine whether to 
cross or not, while in the case of the larger rivers crossing 
may be out of the question. 

When there is a choice of sides, both banks should be 
traversed by the engineer on reconnaissance, and while exam- 


By FIELD ENGINEERING 


ining in detail the one side he should take a general and com- 
prehensive view of the other. Only thus can he gain a complete 
knowledge of either side. ‘The points to be considered are the 
relative value of the property on either side, the number and 
size of tributary streams and probable cost of crossing them, 
the cost of graduation as affected by the amount and character 
of the material to be removed and the lability to land slides, 
the amount and degree of curvature required, and the proba- 
ble revenues which the road can command. IH, in respect to 
these points, one bank of the stream gives the more favorable 
result all the way, the question is decided at once; but in 
case the greater inducements are found on either bank alter- 
nately, as usually happens, the propriety of bridging the 
stream, with the costs and advantages, must be considered as 
an additional element in the problem. 

4. When no water-course offers along which the road may 
be located, the difficulties of selecting a route are increased, 
and these usually become greatest when the streams are found 
to run about at right angles to the direction of the road. Val- 
leys and ridges are to be crossed alternately, involving the 
necessity of ascending and descending grades, diverting the 
road from a straight line, and increasing the distance and cury- 
ature. The engineer must now seek the lowest points on the 
ridges, and the highest banks at the stream crossings, in order 
to reduce as much as possible the total rise and fall, but these 
points must be so chosen relatively to each other as to admit 
of their being connected by a grade not exceeding the maxi- 
mum which may be allowable. The intervening country 
between summit and stream must usually be carefully exam- 
ined, even on reconnaissance, to determine where the assumed 
grade will find sustaining ground at a reasonable expense for 
graduation and right of way. 

In selecting stream crossings, regard should be had not only 
to the height of the bank, but also to the character of the bot- 
tom, its suitability for foundations, and its liability to be 
washed by the current. The direction and force of the cur- 
rent should be observed, and its behavior during freshets, and 
the extremes of high and low water ascertained, if possible. 
An approximate estimate of the cost of bridging may be made. 
- §,. The engineer should not only seek the best ground on the 
route first assumed, but should have an eye to all other possi- 


RECONNAISSANCE ~ See 


ble routes, holding them in consideration pending this accu- 
mulation of evidence, and being ready, finally, to adopt that 
one which promises the-greatest ultimate economy. He 
should be able to read the face of the country like a map, 
and to carry in his mind a continuous idea or image of any 
line he is examining, so as to judge with tolerable accuracy of 
the influence any one portion of the line may have on another. 
as to alinement and grade, even though many miles apart. 
In the successful prosecution of a reconnaissance he must 
depend mainly on his own natural tact and a judgment 
matured by experience. 

6. The engineer will bring to his aid in the first place the 
most reliable maps, and those drawn on the largest scale. 
The sectional maps of United States surveys will be found 
very useful when they exist. In addition to these it is often 
* desirable to prepare a map on a scale of one or two inches to 
a mile, on which will be drawn the principal features of the 
country to be traversed, such as streams, roads, towns, and 
the principal ridges, if known, but leaving the further details 
to be filled in by the engineer as he progresses. Such a map 
furnishes a correct scale for his sketches, and saves much 
valuable time, as he has only to sketch what the map does 
not contain, and occasionally to make corrections when he 
finds the map to be in error. He also notes on the map the 
governing points of the route, such as the best crossings of 
streams, ridges, or other roads, and any point where the line 
will evidently be compelled to pass. He may then indicate 
the route by a dotted line on the map drawn through the 
governing points. Having traversed the route in one direc- 
tion he should retrace his steps, verifying or correcting his 
observations, and making such further notes as seem impor- 
tant. When in a densely wooded country, with but few 
openings, it may be impossible for him to get a commanding 
view from any point that will afford him the necessary 
information as to the general topography. He must then 
depend largely upon instrumental observations, taking these 
more frequently, and noting carefully all details likely to 
prove useful in future surveys. 

7. The instruments required on an extended recon- 
naissance are the barometer and thermometer, the hand or 
Locke level, a pocket or prismatic compass, and a telescope or 


4 FIELD ENGINEERING 


strong field-glass. To these may be added a telemeter for 
measuring distances at. sight, but when good maps are to be 
had this instrument is seldom needed. So also some portable 
astronomical instruments are necessary in a new country, 
for determining latitude and longitude, but would only be a 
useless incumbrance in a settled district. 

8. The mercurial barometer has generally been relied upon 
for the determination of heights, but owing to its inconvenient 
dimensions and the danger of breaking, it is now discarded 
by railroad engineers in favor of the more portable aneroid 
barometer, except in the case of trans-continental surveys. 
and when astronomical instruments are to be used also. 

9. The best aneroids are designed to be self compen- 
sating for temperature, so that with a constant atmospheric 
pressure the reading shall be the same at all temperatures of 
the instrument. This, however, being a very delicate adjust- 
ment, is not always successfully made, so that each instrument 
is liable to have a small error due to temperature peculiar to 
itself. This error will be found rarely to exceed one hundredth 
of an inch, plus or minus, per change of 10° Fahr., and is fre- 
quently much less than this. Just what the error is in a 
particular instrument may be determined by careful com- 
parison with a standard mercurial barometer at the extremes 
of temperature, assuming the error found as proportional to 
the difference of temperature for all intermediate degrees of 
heat. The error having been determined for any aneroid, it 
should be applied, with its proper sign, to every reading to — 
obtain the true reading. 

The sizes generally used are 12 and 2} inches in diameter, 
respectively, and experience seems to prove that there is no 
advantage in using larger sizes, but rather ‘the contrary. 

10. The ordinary barometric formulas and tables have 
been prepared with reference to the mercurial barometer. In 
order that they may apply to the aneroid, it is necessary that 
the latter should be adjusted to read inches of mercury iden- 
tically with the mercurial column at the sea level at a tem- 
perature of 32° Fahr. But as the aneroid, unlike the mercurial 
column, requires no correction for latitude, nor for the vari- 
ation in the force of gravity due to elevation, that portion of 
the formula which provides for such corrections, as well as that 
which provides for a correction due to the temperature of the 


RECONNAISSANCE 5 


instrument itself, may be omitted when using an aneroid. 
Thus the general formula is very much simplified, and be- 
comes 
bee hs . ty + 4! + 64° 

2 = log i 60384.3 (1 + S00 ) 


in which h; and h’ are the readings of the aneroid in inches, 
and ¢, and t’ the readings of a Fahrenheit thermometer at the 
lower and upper of any two stations respectively, and z is the 
difference in elevation in English feet of those stations. 

To facilitate the. calculation of heights by this formula, we 
may write 


Log M69384.3 = [log h; — log h’| 60384.3 


and since only the difference of the logs is required, this will 
~ not be affected, if we subtract unity from each. The quan- 
tities in Table XV are prepared, therefore, by the formula 


(log h — 1) 60384.3 


for every 725 of an inch from 19 inches to 31 inches. 


t, +t’ — 64° for d 
——9007 for every degree 


of (t; + ¢’) from 20° to 200° Fahr. 

11. To find the difference in elevation of any two stations by 
the tables: 

Take the difference of the quantities corresponding to h; and 
h’ in Table XV as an approximation, and for a correction 
multiply this difference by the coefficient corresponding to 
(4 +t), in Table XVI, adding or subtracting the product 
according to the sign of the coefficient. 


Table XVI contains values of 


Ezample.— 
Lower Sta. Upper Sta. 
in, in, 
Aneroid hy, = 29.92 fiiicarQarny 
Thermometer ft, = 77°.6 t’ = 70°.4 
By Table XV for 29.92 we have 28741 
for 23.57 22485 


Difference 6256 
By Table XVI for 77.6 + 70.4 = 148 we have + 0933 
Then 6256 X .0933 = 583.6848 
and »s 6256 + 584 = 6840 ft. = z2.—Ans. 


I 


6 FIELD ENGINEERING 


12. Certain precautions are to be observed in the use of the 
aneroid. When the index has been adjusted to a correct 
reading by means of the screw at its back, it should not be 
meddled with until it can again be compared with a standard 
mercurial barometer, and even then some engineers prefer to 
take note of its error, if any, rather than disturb the aneroid. 
Again, since the principle of compensation supposes the 
aneroid to have a uniform temperature throughout its parts, 
it must be guarded against sudden changes, as otherwise the 
metallic case will be considerably heated or cooled before the 
change can affect the inner chamber, thus inducing very erro- 
neous results. . The aneroid, therefore, should seldom be taken 
from its leather case, nor exposed to any radiant heat of sun 
or fire, nor worn so near the person as to increase its tempera- 
ture above that of the surrounding atmosphere. If removed 
to an atmosphere of decidedly different temperature, time 
must be allowed for the aneroid to be thoroughly permeated 
by the new degree of heat. The aneroid should be held with 
the face horizontal while being read; it should be handled 
carefully, and all concussions avoided, and it should be com- 
pared with a standard as often as practicable to make sure that 
it has suffered no derangement. Observing these precau- 
tions, and having a really good aneroid, the engineer should 
obtain excellent results in the estimation of heights. It has 
been found that the slight error in compensation, previously 
alluded to, is subject to a change during the first year or two 
after the instrument is made, but subsequently it becomes 
quite permanent. 

13. For the purpose of obtaining approximate elevations 
by a simple inspection of the dial, the modern aneroid is pro- 
vided with a secondary scale reading hundreds of feet, which 
is placed outside the scale of inches. It is divided according 
to the following formula prepared by Prof. Airy: 


babe ti 005 
1000 





Z= 55032 rant 
in which it is evident that no correction for temperature is 
required when the average temperature of the two stations is 
50°. When the two scales are engraved on the same plate the 
zero of the scale of feet is coincident with 31 on the scale of 
inches; but in some aneroids the scales are on two concentric 


ie mt 


RECONNAISSANCE . 7 


plates, so that the zero of one may be made to coincide with 
any division of the other, which is in some respects an ad- 
vantage. 

14. The theory of the barometer, as expressed in the above 
formulas, assumes the atmosphere to be at rest, and its pres- 
sure affected only by temperature, whereas, in fact, the pres- 
sure at any point is liable to sudden changes due to variations 
in the force of the wind, the amount of humidity, ete. The 
best way to eliminate errors due to these causes is to take read- 
ings simultaneously at the points the elevations of which are 
to be compared. For this purpose an assistant should be 
stationed at some point of known elevation contiguous to the 
route to be surveyed, and provided with an aneroid similar to 
that carried by the engineer. The aneroids, time-pieces, and 
thermometers having been compared at this point, the assist- 
‘ant should record the readings every ten minutes, with the 
time, temperature, and state of the weather. The engineer 
will thus have a standard with which to compare his own 
observations. If the survey is so extended that the same con- 
ditions of atmosphere are not likely to be experienced by the 
two observers, the assistant should be instructed to move for- 
ward to a new station at a designated time; or two assistants 
may be employed, one at each of two stations between 
_ which the engineer intends to make a reconnaissance. Even 
with these precautions no attempt should be made to obtain 
the elevation of important points during, or just before, or 
after a storm of wind or rain. 

15. When but one aneroid is used the observations at the 
several stations should be taken as nearly together as possible 
in point of time, and then repeated in inverse order, taking 
the mean of the observations at each station, and repeating 
the whole operation if necessary. Only approximate results 
can be hoped for, however, with a single instrument, unless 
the atmospheric conditions are very favorable. 

16. The Locke Level is an instrument in which the 
bubble and the observed object may be seen at the same 
instant, enabling the operator to keep the instrument hori- 
zontal, while holding it in the hand, like an ordinary spy-glass. 
While quite portable, it enables the observer to define rapidly 
all visible points of the same elevation as his own, and to 
estimate from *these the relative heights of other points. It 


8 FIELD ENGINEERING 


may be made useful in a variety of ways which easily suggest 
themselves to the engineer in cases where no great precision 
is required, and where a more elaborate level is not at 
hand. \ 

17. The Prismatic Compass is a portable instrument 
with folding sights, in using which the bearing to an object 
may be read at the same instant that the object is observed. 
The bearings are read upon a floating card, graduated and 
numbered from zero to 360°, so that no error can be made in 
substituting one quadrant for another. ‘The instrument may 
be held freely in the hand during an observation, though better 
results are obtained by giving it a firm rest. 


CHAPTER II 


PRELIMINARY ‘SURVEY 


18. A preliminary survey consists in an instrumental exam- 
ination of the country along the proposed route, for the 
purpose of obtaining such details of distances, elevations, 
topography, etc., as may be necessary to prepare a map and 
profile of the route, make an approximate estimate of the cost 
of constructing the road, and furnish the data from which to 
definitely locate the line should the route be adopted. The 
’ survey is more or less elaborate, according to circumstances. 
In case the country is new, or the reconnaissance has been 
incomplete, or if several routes seem to offer almost equal 
inducements, the survey will partake somewhat of the nature 
of a reconnaissance, and will be made more hastily than if but 
one route is to be examined, and that, perhaps, presenting 
serious engineering difficulties. The survey is made as expe- 
ditiously as possible, consistent with general accuracy, but 
should not usually be delayed for the sake of precision in 
matters of minor detail. 

19. For preliminary survey the Corps of engineers is 
organized as follows: 

A locating engineer, a transitman, two chainmen, one or 
two axemen, a stakeman, and a topographer, these forming 
the transit (or compass) party, to which a flagman is some- 
times added; a leveler and one or two rodmen, forming the 
level party; and to these is sometimes added two or three 
rodmen to assist the topographer. 

20. The locating engineer takes charge of the corps, 
and directs the survey. He ascertains or estimates the value 
of the lands passed over, the owners’ names, and the boundary 
lines crossed by the line of survey. He examines all streams, 
and estimates the size and character of the culverts and 
bridges which they will require; he notices existing bridges, 
and inquires concerning their liability to be carried away by 

vs 9 


10 FIELD ENGINEERING 


freshet; he selects suitable sites for bridges, examines the 
character of the foundations, the direction of the current rela- 
tively to that of the line, and considers any probable change 
in the direction of the current during freshets; he inspects the 
various soils, rocks, and kinds of timber as they are met with 
and takes full notes of all these and kindred items in his field 
book. He not unfrequently assumes in addition the duties of 
topographer. He should run his line as nearly as may be over 
the ground likely to be chosen for location, so that the infor- 
mation obtained may be pertinent, and so that the length of 
the line, the shape of the profile, and the estimate based on 
the survey may approximate to those of the proposed location. 
To this end he has due regard to the levels taken, and when 
they show that the line as run fails to be consistent with 
allowable grades, he either orders the corps back to some 
proper point to begin a new line, or makes an offset at once 
to the better position, or continues the same line with some 
deflection, simply noting the position and probable elevation 
of better ground, as in his judgment he thinks best. He 
should at all times maintain a friendly attitude toward pro- 
prietors, and by his polite bearing endeavor to secure their 
cordial support of the new enterprise. If he is tolerably cer- 
tain that the location will follow nearly the line of the prelim- 
inary survey, he should have with him some blank deeds of 
right of way, and let these be signed by land-owners while 
they are favorably disposed. When this cannot be done, a 
blank form of agreement. to allow the surveys and construc- 
tion of the road to proceed until such time as the terms of 
right of way may be agreed upon may be made very useful. 
He also selects quarters for his men, and in case of camping 
out he directs the movements of the camp equipage. 

21. The transitman takes the bearings of the courses 
run, and makes a record of them, with their lengths, or the 
numbers of the stations where they terminate. He sees that 
the axemen keep in line while clearing, and the chainmen 
while measuring; he takes the bearings of the principal roads 
and streams, and of property lines when met with. In an 
open country he may save time by selecting some prominent 
distant object toward which the chainmen measure without 
his assistance, while he goes forward and prepares to take the 
direction of the course beyond. In traversing a forest with 


PRELIMINARY SURVEY 11 


not too dense undergrowth, when the line is being run to suit 
the ground according to a given grade, it is a good plan for the 
transitman to go ahead of the chainmen as far as he can be 
seen, select his ground, take his direction by backsight on the 
last station, and then have the chainmen measure toward 
him. In this case both he and the head chainman should be 
provided with a good-sized red and white flag, mounted on a 
straight pole, to be waved at first to call attention, and after- 
ward held vertically for alinement. Otherwise a flagman 
must oe added to the party, who will select the ground ahead 
under the instructions of the locating engineer, and toward 
whom the«urvey will proceed in the usual manner. 

22. The head chainman drags the tape and carries a 
flag which is put into line at the end of each chain length by 
the transitman or the rear chainman. It is his duty to know 
‘that this flag is in line and that his tape is straight and hori- 
zontal before making any measurement, and to show the 
stakeman where each stake is to be driven. A stake is usually 
driven at the end of each measured chain length, called a 
station, though in an open and level country the stakes at the 
odd stations may be omitted, in which case marking pins are 
used to indicate the odd stations temporarily. In case there 
is much clearing to be done the head chainman plants his flag 
in line, and ranging past it, indicates to the axemen what is 
to be cut, going a little in advance through the bushes so that 
they may work toward him. The head chainman should be a 
quick, active and strong man, with a good eye and a taste for 
his work, as very much of the real progress of the survey 
depends upon him. 

23. The rear chainman holds his end of the tape firmly 
at the last stake or pin by his own strength, not by means of 
the stake. He keeps the tally by the pins when they are used, 
and watches the numbers on the stakes to see that they are 
correct. The end of a course should always be chosen at the 
end of a chain, if possible, and if not, then at a 10-foot mark, 
as thus the labor of plotting the map will be much lessened. 
The numbering of stations is not recommenced with each 
new course, but is continued from the beginning to the end 
of the survey, through all its courses, and if one course ends 
with a portion of a chain the next course begins with the 
remainder of it.. It is the rear chainman’s duty to attend to 


12 FIELD ENGINEERING 


this, holding the proper foot mark at the transit station. Any 
fraction of a chain measured on the line is called a plus, and 
is counted in feet from the previous station. The length of 
an offset in the line is never included in the length of the line, 
but if the line should change its course by a right angle, or 
more, or less, the numbering would go on as usual. 

24. The axemen should be accustomed to chopping and 
clearing, and are, therefore, to be selected in the country 
rather than the city. They will cut out so much of the 
underbrush and overhanging branches as may interfere with 
the sight of the transitman or leveler; but care must be taken 
not to cut unnecessarily wide, and no tree of considerable 
size should be felled, except in rare instances. When running 
by compass, if the transitman goes ahead of the chain, he 
can always select a position so that no large tree will 
interfere; or, if the line must be produced and strikes a tree, 
the transit may be brought up and set close to the tree on 
the forward side as nearly in line as can be estimated, the 
slight error in offset being neglected, since the line will be 
produced parallel to itself by the needle. 

25. The stakeman prepares and marks the stakes, and 
drives them at the points indicated by the head chainman. 
When no clearing is needed, the axemen keep him supplied 
with stakes, as the rapid progress of the chain will only give 
him time to drive them. The stakes should be 2 feet long 
and pointed evenly so as to drive straight, and are blazed or 
faced on two opposite sides, one of which is marked in red 
chalk with the number of the station. The stake must be 
driven vertically, and with the marked face to the rear, 
so that it may be read by the rodman as he follows the 
line. 

26. The topographer makes accurate sketches of all 
features of the country immediately on the line, and extends 
the sketches as far each side of the line as he can, in a book 
prepared for the purpose. He must never sketch in advance 
of the chain, nor in advance of his own position. His work 
should be done to scale as nearly as possible, using the same 
scale for distances on the line and at right angles to it. The 
scale adopted should never be less than that of the map to be 
made from the sketches. The ruled lines of a field book 
are usually one quarter of an inch apart, so that the scale 





PRELIMINARY SURVEY 13 


of one line to a station equals about 400 feet to an inch. 
This is the smallest scale ever used. The scale of two lines to 
a station is most convenient for general use. Four lines to a 
station are needed in special cases to show details, as in pass- 
ing through villages. The scale may be changed from time to 
time as found necessary, but no two scales should ever be used 
on the same page. ‘The numbering of the stations up the page 
indicates the scale of the sketch. 

27. When the contours of the surface are required, the to- 
pographer may join the level party in order that his estimates 
of heights and slopes may be corrected by the instrument. 
He should never draw a mass of contours indiscriminately, 
but should sketch them as they exist at a uniform vertical 
interval. This interval may be assumed at 5 feet in a 
gently rolling country, and at 20 feet in a mountainous one, 
but an interval of 10 feet will be found most convenient 
generally: If the topographer accompanies the level he can 
assume. the contours at the even tens of feet in elevation, and 
mark them so, noting where a contour crosses the surveyed 
line, and sketching its direction and shape both ways from 
that point. He will-estimate the rate of slope of the ground 
at right angles to the line as so many feet per hundred, and 
record it from time to time, noting ascent from the line on 
either side by “‘A,’’ and descent by “‘ D.’”’ If the slope changes 
within the limit of the page, the line of change may be 
sketched and the next slope recorded. When little banks or 
terraces occur, or bluffs and rocks, which cannot be suf- 
ficiently indicated by contours, they should be shown by 
hatchings, and the height noted. Special care should be 
taken to sketch roads and houses in their correct positions 
and dimensions, the latter to be either measured, paced or 
estimated. The dimensions should also be recorded in num- 
bers. The outline of forests may be shown by a scalloped 
line, and the kind of timber, and whether dense or scattered, 
written within the inclosed space. Correct outlines are essen- 
tial, but no time should be given to shading up a sketch with 
conventional signs. <A single sign, or the name of the thing 
intended, is all sufficient. Land-owners’ and residents’ names 
should be recorded whenever they can be obtained, as well as 
the names of roads, streams and public buildings. 

28. The leveler takes charge of the level party and keeps 


“14 FIELD ENGINEERING 


the notes of his work. He reads the rod on benches and 
at turning points to hundredths of a foot and to tenths at 
other points. He should direct a bench to be made at least 
once every half mile, and in a very rough country every 
quarter of a mile. The benches need not be far from the line, 
and, if well chosen, may be used as turning points, thus saving 
time. The elevation of turning points must be computed 
when taken, so that the elevation of any one of them may be 
instantly given when called for, and the other elevations will 
be filled in as far as may be without delaying the survey. As_ 
the levels are usually the most essential part of the survey, 
much care should be taken to have the instrument well 
adjusted and truly level, and the rod held vertically and 
correctly read on turning points, but the intermediate work 
should not be so done as to delay the party unnecessarily. 
The leveler should use every endeavor to follow closely after 
the surveying party, so that the locating engineer and topog- 
rapher may have the advantage of his notes. 

29. The rodman’s first duty is to hold the rod ver- 
tically, and he must learn to do this in calm or windy weather, 
in level field or on side hill. He may carry a small disk-level, 
which applied to the edge of the rod will show when it is 
vertical. The turning points are. to be selected for firmness 
and definiteness, and so that they will afford a clear view 
from beyond for a backsight. The rod is held for a reading 
on the ground at every stake, the number of which is called 
out to the leveler as soon as the rodman arrives at it; the rod 
is also to be held at every prominent change of slope on the 
line, as the crest and foot of every bank, the rodman calling 
out its distance from the last stake as plus so many feet, but 
all gentle undulations and minor irregularities are to be 
neglected. The rod will always be read at the surface of a 
stream or pond, and also at its deepest part on the line, 
when possible; otherwise the depth of the water may be 
found by sounding, and so recorded. Should the line run 
along a stream the surface will be taken occasionally, opposite 
certain stations, and in case of a canal, the elevation of surface 
above and below each lock must be noted. The rodman 
makes inquiry for high-water marks or seeks traces of them 
himself in an uninhabited district, and holds the rod upon 
them that their elevation may be determined. . The rodman 


PRELIMINARY SURVEY 5 


carries a small axe or hatchet with which to make benches 
and to trim out any stray branch that may intercept the 
leveler’s view. 

30. Where detailed information is desired on a 
narrow stretch of country near the preliminary line, one cr 
more rodmen may assist the topographer in securing this. 
The contours and all important features such as highways, 
fences, buildings, bodies of water, etc., should be located. A 
contour may be located either by hand level, level rod and 
tape, or by ‘horizontal and vertical rods. The first method 
is probably in more general use, but the second has a decided 
advantage in the fact that the work may be done indepen- 
dently of the level work. For further information on these 
two methods see Chapter X VIII on Topographical Sketching. 

31. In defining the duties of the members of the corps, the 
instruments used have been incidentally noticed. 

32. The compass is sometimes used instead of the 
transit on preliminary surveys, because it can be operated 
more rapidly, is lighter, and usually has a better needle. 
It may have either plain sights or telescope, and be mounted 
on tripod or jacob staff. The simpler forms are preferred 
for forest work. Not unfrequently the engineer’s transit 
is employed, but using the needle. When local attraction 
is found to exist: to such an extent as to destroy confidence 
in the needle, or when obstacles are very great, the transit 
should always be used. In this case, the preliminary line 
should be run by backsights and deflections. The survey 
then partakes of the nature of a location, and should be 
conducted with similar care and fidelity. 

33: The engineer’s chain which was formerly in general 
use, is 100 feet long and composed of one hundred links. It 
has largely been superceded by the steel tape, although it is 
occasionally made use of. The word “ chain ”’ is a convenient 
term, meaning 100 feet, and is commonly used. A set of 
marking pins will be found of service for temporary stations 
in running an offset line to some object of importance, or as 
markers in chaining on a hillside too steep for a whole chain. 
One or more plumb bobs, each having several yards of 
small cord wound on a carpenter’s spool, should be provided. 

34. The axes should be of best quality, with hand-made 
handles, and nottoo heavy. The axe of the stakeman should 


16 FIELD ENGINEERING 


have a fine edge for dressing and a broad head for driving 
the stakes. When the stakes are not required to be over two 
feet long, a stout basket, having a square, flat bottom, 2614 
inches, should be furnished the stakeman. He will then pre- 
pare a basketful of stakes, ready marked, and place them in 
the basket regularly, in the reverse order of their numbers, so 
that they will come to hand as wanted. A small hand-saw 
no larger than the basket, with rather coarse teeth, wide set, 
will be found extremely useful in cutting stakes with square 
heads and of uniform length, and much more rapidly than can 
be done with an axe. When not in use, it is to be strapped to 
the inside of the basket, to prevent its being lost by the way. 
When the basket is about empty, the stakeman, with the 
assistance of the axemen, can soon replenish it, and the stakes 
being all numbered at once, there is less danger of a mistake 
being made in the tally than when they are marked only as 
wanted. 

35. The level should be the regular engineer's level, the 
same as used on location. 

36. The rod should be self-reading, t.e., to be read by 
the leveler, as too much time would be consumed in the con- 
stant adjustment of a target by the rodman. It should be 
as long as can be conveniently handled in order to reduce the 
number of turning points on hill sides. A-very convenient 
rod may be made of thoroughly seasoned clear white pine, 
16 feet long and 2 inches wide, with a thickness of 1 inch at 
the bottom, increasing to one and a quarter inches at 6 feet 
from the bottom, and then gradually diminishing to three 
eighths of an inch at the top. The rod is shod with a 
stout strap of steel, extending 5 inches up the edges; and 
secured by screws. The top is protected for a few inches by 
a plate of sheet brass on the back. The face of the rod is a 
plain surface throughout, and is graduated from the lower 
edge of the steel shoe as zero. The divisions are fine cuts 
made with the point of a knife. At the foot and half-foot 
points the cuts extend across the face. For the tenths and 
half tenths they extend three quarters of an inch from the 
right-hand edge, terminating in a line scribed parallel to the 
edge of the rod, thus forming rectangular blocks half a tenth 
wide, every other one of which is painted black, the body 
of the rod being white. The feet are indicated by numerals 


PRELIMINARY SURVEY hw 


painted red on the blank part of the face, each figure standing 
exactly on its foot mark, and being exactly one tenth high. 
For the figure 5 the Roman V is substituted and for 9 the 
Roman IX, so that in case a dumpy level is used the 5 may 
not be mistaken for a 3, nor the 9 fora 6. At the half-foot 
points a red diamond is painted, so that the graduated line 
bisects it. No other figures nor graduations are required. 
With this rod the leveler can read quite accurately to hun- 
dredths of a foot, and after some practice can estimate the 
half hundéedths. 

37. The horizontal rod for cross-levels may be made 
of white pine, 10 feet long and 1 inch thick by 3 wide, tipped 
with brass, painted white, and graduated to feet and tenths. 
It must be a straight edge, and is leveled by a pocket level 
placed upon it when needed, or by a small level embedded 
permanently in one edge. ‘The vertical rod to be used with it 
is made of pine 8 feet long and one and a quarter inches 
square, and graduated to feet, tenths, and half tenths. All 
rods when not in use should be laid on a flat surface to pre- 
vent their being sprung. Leaning them in a corner soon ruins 
them for use. 

38. The clinometer is any small instrument which will 
measure the slope angle of the surface. The angle is always 
estimated from the horizon, a vertical being 90°. The rise 
per 100 feet is found by multiplying the nat. tangent of the 
slope angle by 100. It may often be found more easily by the 
leveler reading the rod at a station and then 100 feet left or 
right of the line. If surface measures are taken in connec- 
tion with a slope angle they are reduced to horizontal meas- 
ures by multiplying them by the cosine of the slope angle. 

39. The plane-table is rarely if ever used on prelimini- 
nary surveys in the United States. Occasional bearings taken 
to prominent objects by the transit man, or the use of a 
prismatic compass by the topographer in connection with his 
sketches, is found to answer every purpose. 

40. In a survey made with transit deflections, it 
is necessary to add a back flagman to the party, who will 
hold his flag or rod on the point last occupied by the transit, 
so that the transitman may take a backsight upon it. The 
direction of a new course in each case is determined by the 
deflection angle to the right or left of the preceding course 


18 FIELD ENGINEERING 


produced. The bearing of one long course near the beginning 
of the survey having been carefully ascertained, the bearing 
of each succeeding course is calculated from the deflections, 
and entered in a column of the field book headed Calculated 
Bearings, from which the line is afterwards plotted. The 
magnetic bearing of each course should also be taken from 
the needle, and recorded as such, but is used only as a check 
on the transit work. The deflections should be made in 
degrees, halves, or quarters, if possible, to facilitate the cal- 
culation of the bearings, and to admit of using a traverse 
table. 

41. The attached level and vertical arc of the transit are 
useful in determining approximately the grade of the line run 
in advance of the level party, or in seeking for one assumed 
grade to which it is desired that the line shall conform. For 
this purpose it is only necessary to set the vertical are to the 
angle corresponding to the grade as given in Table XIV, and 
let the head chainman move about until a point on his rod at 
the same height from the ground.as the telescope is covered 
by the horizontal cross-hair. 

42. The point on the ground where a transit is set up is 
marked by a good-sized plug, flat headed, and driven down 
flush with the ground, with a tack set in the head to show the 
exact point or center. This is called a transit point. When 
a transit point occurs at a regular station, the stake bearing 
the number of that station is set 3 feet to the left of the line 
opposite the plug and facing it. When a transit point occurs 
between stations the stake is driven 3 feet tothe left of it, 
marked with the number of the preceding station + the 
distance from that station in feet. 

43. As a transit is capable of giving a line with great pre- 
cision, it is important that the flags used in connection with 
it should be equally precise in giving points. An excellent 
flag for this purpose is made of well-seasoned clear white 
pine 10 feet long, two and a half inches wide, and 1 inch 
thick. It is tapered for the last 4 inches to an edge at one 
end, the edge being formed: at the middle of the width. The 
tapered end is shod with a band of steel covering the edge of 
the rod, and secured by screws, and the steel is brought to 
a sharp edge at the point of the rod. The rod is then painted 
white and tipped with brass at the square or upper end. A 


PRELIMINARY SURVEY 19 


center line on the face is then struck from the point of the steel 
to the middle of the brass tip by means of a piece of sewing 
silk, and a fine cut made with a knife and steel straight edge. 
The center line must not be scribed parallel to one edge of the 
rod, as this is rarely ever straight. The face of the rod is 
then divided into 1-foot spaces, measured from the head of 
the rod, and these are painted red on either side of the center 
line in alternate blocks. On the back of the rod at three and 
a half feet from the point is placed a small ground-glass 
bubble-tube, mounted very simply, and attached to the rod by 
a brass plate and screws, and guarded by two blocks of wood 
for protection. The center line of the rod is made vertical by 
a plumb-line while the level tube is being attached, which ever 
after secures a vertical rod. If only 2 feet of this rod can 
be seen over any obstruction, a point can be set with great 
’ precision, provided the level tube is in adjustment. This 
flag can also be used as a plumb in chaining with much more 
satisfaction than a cord and weight, especially in windy 
weather. 

44. A transit survey usually requires more clearing than 
one made by. compass. When a given course is to be pro- 
duced in a forest, some large trees will inevitably be encoun- 
tered, but the labor and delay of felling them may be avoided 
by the use ot auxiliary lines. These may be classified as 
running parallel to the main line, at a small angle with it, or 
at a large angle with it. 

45. The parallel line is established by means of two 
short perpendicular offsets measured with care before reach- 
ing the obstacle, and the main line is established beyond the 
obstacle by means of two more equal offsets. But since short 
backsights are to be avoided, these offsets should be at least 
100 feet apart, and it may be difficult to find a parallel line 
of sufficient length which does not strike some other obstacle, 
or at least require considerable extra clearing. 

46. The auxiliary lines making a small angle with 
the main line are more convenient, not only on this account, — 
but because they require a less number of transit points. By 
them an isosceles triangle is formed on the ground, having 
the intercepted portion of the main line as base, and the 
vertex near the obstacle. The deflections at the points where 
the lines leave,and join the main line are similar and equal, 


20 FIELD ENGINEERING 


and the deflection at the vertex is double in amount and oppo- 
site in direction. No calculation is necessary, for the error 
in measurement due to the deviation is too small to be noticed, 
and since the main line is immediately resumed, the calculated 
bearings of the auxiliary lines are unnecessary. Should the 
point where the second line joins the main line prove unsuit- 
able for a. transit point, the second line may be produced to 
any convenient point beyond, and so go to form an isosceles 
triangle on the opposite side of the main line, the triangle 
being completed by running a third line parallel to the first, 
and equal to the difference of the first and second. Again, 
the second line may encounter a serious obstacle before reach- 
ing the main line. To avoid this a parallel to the main line 
may be run from the end of the first line for a convenient 
distance, and there the second line be put in parallel and equal 
to its first position, as before described, thus forming a 
trapezoid. 

47. The following general solution of this problem 
allows the engineer to make use of any number of auxiliary 
lines, provided that none of them make an angle of much 
more than one degree with the main line, with a certainty 
of resuming the main line in position and direction at the 
extremity of any course desired, and without necessitating 
any trigonometrical calculation. It is based on the assump- 
tion, practically true for small angles, that the sines are pro- 
portional to their angles, and is expressed by the following 
rule: . 

Call all deflections to the right plus, all to the left minus; 
multiply the length of each course in feet by the algebraic 
sum in minutes of all the auxiliary deflections made to reach 
that course; take the algebraic sum of these products, and 
when the sum equals zero the extremity of the last course 
will be on the main line. The deflection required at that 
point to give the direction of the main line is equal to the 
algebraic sum of all the preceding deflections, but taken with 
the contrary sign. 

Thus, if we have left the main line at A, and run by these 
notes: (Fig. 1.) 


PRELIMINARY SURVEY 21 





Sta. Defl. Dist. Factors. Products. 
A iste 8 190 2 ¥ + 16x 190.= + 3040 
B Shel, 120 ac — 15 X 120 = — 1800 
C 18’R 175 ae + 3175 = + 525 
D Ae 265 Ze — 10 X 265 = — 2650 
E 15’R Fa + 5 X (?) 

3565 — 4450 
and their algebraic sum is — 885 
Therefore to render the sum zero we must add p 


885 as the product of the last course. But 5/ is 
already given as one factor, so that the other 


2Q 
factor must be = = 177, which is the length of E 


the last course, giving some point /’ on the main 
line. “The deflection at / from the last course 
to give the direction of the main line is 


16 — 31+ 18 = 13 +4 15 = 5’ 


and changing the sign we have —5’; that is the 
deflection is to the left. 

The distance on the main line from A to 7 
equals the sum of the courses, or 927 feet, but 
this we have by the stations, which have been 
kept by stakes in the ordinary way. All the 
stakes on the auxiliary lines will be more or less 
off the main line, but as these offsets are usually 
very small, they are considered of no con- 
sequence on a preliminary survey through a 
forest. In Fig. 1 the offsets are very much magnified. The 
field notes of such auxiliary courses should be kept, not as 
regular notes, but on the margin or opposite page, and in such 
a way that, while the line may be retraced by them on the 
ground, the draughtsman may see that it is not necessary to 
plot them, when a straight line ruled and measured through 
is sufficient. It is obvious that in selecting a closing course 
either the deflection may be assumed and the length calcu- 
lated, or vice versa, but care should be taken to assume such 
values as do not invoive a fraction in either factor, if possible. 

48. The method of passing an obstacle on the line by 


ErGes I 


22 FIELD ‘ENGINEERING 


auxiliary lines at a large angle with the main line will 
be resorted to only when circumstances are such that the 
other methods mentioned cannot be employed, as in passing 
a building, pond, or densely wooded swamp. In such a case 
we may turn a right angle with the transit, and measure 
accurately one offset, putting a transit point at its extremity, 
where another right angle will give a parallel line. If the 
offset prove too short for an accurate backsight, a temporary 
point at a sufficient distance may be established for that pur- 
pose on the offset line produced before the instrument is 
removed from the main line. If any other angle than 90° 
is used it should be selected, when circumstances permit, so 
that the distances on the intercepted part of the main line 
may be in some simple ratio to the distances measured 
on the auxiliary line. Thus a deflection of 60° gives a dis- 
tance on the main line equal to half the length of the auxiliary 
course, that is, 


60° gives a ratio of } = 0.5 
Dose Vt cline! 0.6 nearly 
45° 343’ (a9 bc c¢ 0.7 
36° yn 73 73 iz 0.8 
25° 502’ 6c 7: ‘ 0.9 


the angles being taken to the nearest half minute. 

49. If it be desired that the stakes on the auxiliary 
line should stand on perpendiculars through the true 
stations on the main line, a certain correction must be added 
to each chain length depending on the angle which the 
auxiliary makes with the main line. If there is a fraction of 
the chain at either end of the course, a proportional addition 
must be made for this. Thus, by referring to the table of 
external secants, we find that we must add a correction as 
follows: 


2° 333’. ..0.1 ft. per chain 6° 453’. ..0.7 ft. per chain. 


Bags WOE j THIS BOSS 3 
4° 26’ ...0.3 “ if 7°7807/ GOLeSISi * 
Bailie . O41 : 8°. 04ti ah Sots, 14 
543400255" 3 SMO steganl| bias & c 


Gres. 20:6-* £ LL? 2207 ee2ioal an 2" 


PRELIMINARY SURVEY 223 


These methods of suiting the angle to an even measure are 
much superior to assuming an even number of degrees deflec- 
tion, and then calculating the distance by trigonometry. The 
last table, which may be extended indefinitely by reference to 
the table of Ex. secants, is perfectly adapted to chaining by 
surface measure on regular slopes when the slope angle is 
known, the chain being lengthened by the correction corre- 
sponding to the slope angle. 

50. If the chain is lengthened as per above table on auxil- 
iary lines, the numbering of the stakes goes on as usual, but 
they should have an additional mark as X to show that they 
are off the main line; and they may stand facing the true 
stations which they represent, and the length of offset, if 
known, may also be recorded on them. ‘The leveler will then 
understand that he is to read the rod not only at the stakes as 
they stand, but also at the true stations, as nearly as may be. 
The transit man will always make a diagram in his field 
book, showing exactly the method pursued in reference to 
auxiliary lines. Having passed the obstacle, it is advisable 
to return to the main line by a course equal in length to the 
first auxiliary, and making an equal angle with the main line. 
If this cannot be done from the end of the first course, a 
parallel to the main line may be run any convenient distance, 
and the return line then put in, forming a trapezoid. 

51. When there is no obstruction to sight on the main 

line, but only to measurement, a transit point should be 
set in line beyond the obstacle before the transit leaves the 
‘main line, as a check on the other operations, and the main 
line should be afterward produced from this point by back- 
sight on the main line, rather than by deflection from an 
auxiliary line. 

52. The main line should always be resumed as soon as 
practicable, making the auxiliary lines the mere exception. 
When a number of courses at a large angle are likely to be 
required before the main line can again be reached, it may be 
better to consider these as regular courses of the survey, and 
to note them as such. ‘The simplest method is always the best, 
because least likely to involve mistakes. 

53. When the natural obstacles are so numerous and 
of such magnitude as to render any continuous line of sur- 
vey or location extremely difficult, if not impossible, as in the 


24 FIELD ENGINEERING 


case of a bold rocky shore, all the data necessary to a location 
should be gathered with precision on the preliminary survey, 
the measurements and angles being taken with the greatest 
care, and as many checks as possible should be introduced to 
verify the work. In meandering such a shore it is probable 
that a large number of short courses will be used which may 
be measured correctly, but there is liability to error in the 
angles. To verify the latter the more conspicuous transit 
stations on the prominent points of the shore are selected, and 
these being named by the letters of the alphabet, the deflec- 
tions between them are taken by careful observations re-. 
peated a number of times, as for a triangulation. These 
points, joined by tie-lines, then form a survey of themselves, 
much simpler than the full traverse. To obtain the length 
of these tie-lines, the angles between them and the courses 
meeting at the same station are measured. Then since each 
tie-line forms the closing side of a field, in which all the bear- 
ings are known, and all the distances, save one, that one may 
be calculated by latitude and departures. But the angles 
should first be tested for error in each complete field, and if the 
error be large the angles must all be remeasured until the 
error is found and corrected, but if very small it may be dis- 
_ tributed among the angles, or among those most probably 
inaccurate. Before calculating the traverse of any of these 
fields, it will be advantageous to assume, for an artificial 
meridian, a line parallel to the average direction of the shore 
for several miles, and to refer all courses to this meridian for 
their bearing. This meridian is called the azis of the survey, 
and all bearings referred to it are called axial bearings, as 
distinguished from magnetic bearings. The magnetic bear- 
ing of the axis should be some exact number of degrees, so as 
to facilitate the reduction from one system to the other. 

54. In plotting the map, the axis is first laid down, and 
then the lettered stations in their respective positions, after 
which the meandering surveys can be filled in. The map 
being drawn on a scale of 100 feet to an inch, and the 
contours constructed from the notes of the level and cross- 
level parties, the engineer may project the location upon 
it with great certainty and economy of result. But he should 
calculate the traverse of the location as projected, and com- 
pare it with the traverse of the preliminary, to eliminate all 


PRELIMINARY SURVEY 20 


errors in drafting, before taking his notes to the field to repro- 
duce the location on the ground. Any point where the 
location crosses the preliminary should have the same latitude 
and longitude by the traverse of either line. This system, 
though laborious, is the only one that will ensure a successful 
location under the circumstances supposed. Advantage may 
sometimes be taken of cold weather to cross bays and inlets 
on the ice, but there is great liability to error in angles taken 
upon the ice, due both to its motion and to the sinking of the 
feet of the tripod into the ice as soon as exposed to the rays 
_ of the sun. 


CHAPTER III 


THEORY OF MAXIMUM ECONOMY IN GRADES 
AND CURVES 


55. Choice of Routes. Before commencing the field 
work of location it devolves upon the engineer to decide 
as to which of the surveyed routes shall be adopted as being 
most advantageous in all respects, and also to establish the 
maximum grade in each direction and the minimum radius 
of curve on that route. : 

The general considerations which guide the engineer in the 
selection of one of several routes for location are such as 
were hinted at in the chapter on reconnaissance, but upon 
the completion of the preliminary surveys he has at hand 
a large amount of information which enables him to consider 
this important question much more in detail. Unless his 
instructions are explicitly to the contrary, he may assume 
it to be his duty to find the best line, or that one which, for 
a series of years following the completion of the road, will 
require the least annual expense, including interest on first 
cost. The finances of the company may be so limited as 
not to permit the construction of the best line at once, and 
it may then be the duty of the engineer to select the cheapest 
line, or that of least first cost, as a temporary expedient, 
with the expectation of building the road at its best when 
the improved credit of the company will permit. But 
generally he will be able to build the cheaper portions of 
the best line at once, only making deviations and introducing 
heavier grades at the expensive points to avoid a cost beyond 
the present means at his command. The selection of the 
best line may be a question as between different routes or 
as between different grades and curves on the same route. 
We will consider the latter case first. 

56. To solve the problem of true economy we must 
determine the actual expense both of building and operating 

26 


‘MAXIMUM ECONOMY IN GRADES, ETC. DF 


the line at a given maximum grade, and also what changes 
will be made in these expenses by a change in that maximum. 
We have then, on one hand, the annual interest upon the 
original cost, and, on the other, the annual expense of operat- 
ing the road. The best grade is that which will render the sum 
of these two a minimum. Both forms of expense consist 
of two parts: one that is affected by a change in grade, 
and the other that is not. Clearly the former is the only 
one we have to consider in either, since when the sum of the 
variable portions is a minimum, the sum total will be a mini- 
mum also. The varying portions then are functions of the 
grade, though independent of each other. If, therefore, 
we let 2’ represent the maximum grade in feet per mile, 
and let x represent the corresponding value of that portion 
_of the annual expense which varies with the grade, and 
establish the relation existing between the two, we shall 

have x =f (2’). Similarly if we let y represent the interest 
on so much of the first cost as is affected by grade, we shall 
have y =f’ (2’). The problem then is to find that value 
of 2’ which shall render 


ety =amnmum. 


Tet us now seek the complete expression represented by 
ioe Cab 

The clements that enter into this expression are numerous, 
and will be considered in succession. 

57. The tractive force of a locomotive is limited 
by the friction existing between the driving wheels and 
rails, since a locomotive is designed to have its boiler and 
cylinder capacity somewhat in excess. This friction, called 
the adhesion, is expressed as a percentage of the weight 
on the driving wheels, and under ordinary working condi- 
tions is found to be from 20 to 25 per cent of the weight. 
The weight imposed by the driving wheels is limited by the 
ability of the rail material to withstand abrasion and de- 
formation; with iron rails the limit per pair of drivers is 
fixed at 24,000 pounds, with steel rails at 48,000 pounds. 
There are instances of 52,000 pounds permitted. 

58. The expense of running an engine one mile, haul- 
ing a train on the proposed road, can be estimated only from 
the experienceson other roads similarly situated, The items 


28 FIELD ENGINEERING 


of expense while the train is on the road are fuel, the wages 
of the engineer and the train crew, together with many smaller 
items of transportation. 

As previously stated, when lines of different length are to 
be compared for expense per train mile, it is only those items 
affected by grade which concern us in this problem. 

The cost of a train mile has been increasing for some years 
past. An average figure may be $1.50, and the part affected 
by the ruling grade is about 52 per cent of this amount. 

Where the bulk of traffic warrants their use, heavy loco- 
motives give more economical results than lighter ones. 

59. The resistance offered to the motion of a railway 
train is occasioned by a variety of causes, concerning which 
a great deal of uncertainty exists as to their relative effect. 
An investigation which should seek to determine the exact 
amount of each partial resistance, and then by a summation 
derive the total, would be tedious, and, in the present state of 
our knowledge, unsatisfactory. We shall therefore simply 
group the resistances under three general heads, namely: 

Resistance due to uniform motion on a straight, level 
track; 

Resistance due to grade; 

Resistance due to curvature. * 

60. The first of these, considered as an aggregate of 
the various items of friction in locomotive and train, of 
oscillations and impacts, and of resistance of the atmosphere, 
is found to vary nearly or quite as the square of the velocity. 
The friction of a locomotive is greater in proportion to its 
weight than that of a car, owing to its many moving parts, 
so that the resistance of a short train is greater in“propor- 
tion to its total weight than that of a long train. The 
resistance of the atmosphere is greater also in proportion 
to the weight of a short train than of a long one. An empty 
train will offer more resistance in proportion to its weight 
than a loaded one. A formula which shall express the 
resistance of a train to uniform motion must include at least 
the velocity and the weight of the train and locomotive. 

The following empirical formula was given in the former 
editions of this book. It is based upon the records of a large 
number of experiments and is designed to give the resistance 
per ton for all trains, whether freight or passenger, and at 


MAXIMUM ECONOMY IN GRADES, ETC. 29 


any velocity, under ordinary circumstances. Accidental 
circumstances, such as the state of the weather, and the 
condition of the road-bed, rails, and rolling stock, may largely 
modify the resistance, but these, of course, are not taken 
into account in the formula. 


q = 4.82 + (.0053 a 


2 
000478E ) = a 


E+W+T 


where V = the velocity of the train in miles per hour, 2 = 
the weight of the engine and tender in tons, W = the weight 
of cars in tons, T = the weight of freight in tons, and g = 
the resistance to uniform motion in pounds per ton. The 
short ton of 2000 pounds is the one used. 

An elaborate series of experiments * on train resistance 
has been conducted by Prof. E. C. Schmidt of the University 
of Illinois. The tests covered a wide range of velocities, 
weights, and loadings of trains. Formulas were developed 
for several different loadings. The following single empirical 
formula expresses the value of the train resistance in pounds 


per ton: 
ey V + 39.6 — 0.031w 


4.08 + 0.152w 


where w = the average weight of car and load in tons, and 
V and g are as in eq. (1). 

61. The second resistance considered is that due to 
gravity in grades. It varies in the exact ratio of the rise to 
the length of the grade. 

Let Gs = rise of grade in feet per station; 

Gm = rise of grade in feet per mile; 
q’ = resistance in pounds per ton due to grade. 

Then, 


(2) 





q’ = 2000 ae, = 204s 
and e (3) 
: reso tod 1 
Gert pean aE 


The first of these equations expressed in words is easily 
remembered. The grade resistance is equivalent to 20 pounds 
per ton per rate of grade per station. 

* See Bulletins 43 and 59 of the Engineering Experiment Station of 


the University of Illinois; also Journal of the American Society of 
Mechanical Engineers for May and September, 1910. 


30 FIELD ENGINEERING 


62. The third resistance considered is that due to 
curvature of the track. This resistance is due to the friction 
of the wheels upon the top of the rail, and of their flanges 
upon the side of the rail. The top friction is lateral, due 
to the oblique position of the wheel on the rail, and longi- 
tudinal, due to the greater length of the outer rail, since 
both wheels are rigidly attached to the axle. The flange 
friction is due to the reaction of the top friction, which, 
combined with the parallelism of the axles, throws the 
truck into an oblique position on the track. A forward 
flange presses the outer rail, while a rear flange is usually 
in contact with the inner rail. The centrifugal force of the 
car will increase the pressure on the outer rail, unless the 
ties are inclined at an angle sufficient to counterbalance 
this force. But if the ties are inclined too much, or the 
velocity is less, the pressure on the inner rail will be increased. 
An uneven track will cause the truck to pursue a zigzag 
course, increasing the resistance considerably. 

The curve resistance probably varies from about 4 pound 
to 1 pound per ton on a 1° curve. For sharper curves it 
varies as the degree of curve. The value depends on the 
character of the rolling stock, the location of the curve with 
respect to grade, and on the speed of the train. 

63. Curve Compensation. When a ruling grade is 
lessened for curvature, it is said to be compensated for curva- 
ture, or merely compensated, and the equivalent grade resist- 
ance per degree of curve is called the rate of compensation. 

To ascertain the relation between the curve and the grade 
resistance, the following reasoning may be employed: 

Assuming the rate of compensation as 0.035 and that the 
curve resistance varies directly as the degree of curve, we 
have 


(4) 


0.035D = Gs 
and 


i} .847D — Gin 


For definition of degree of curve, see § 84. 

Then if q’’ = the curve resistance in pounds per ton on 
any curve, by combining first eqs. (38) and (4) and sub- 
stituting q’’ for q’ there results 


q = 0,70D © (5) 


x 
MAXIMUM ECONOMY IN GRADES, ETC. 31 


The coefficient of D in eq. (4) is adopted as an average 
of values suggested in the recommendations quoted below. 

Compare also the latter part of § 62. 

Example. If the ruling grade of a road is 1 per cent, 
what is the allowable grade on a 4° curve? 

From first eq. (4) the additional resistance caused by: 
the 4° curve is equivalent to 0.14 of a foot per station. Con- 
sequently the 1.0 per cent grade must be lowered to 0.86 
per cent. 

The American Railway Engineering Association has made 
the following recommendations for the rate of compensation: 

(a) Compensate 0.03 per degree: 

When the length of curve is less than half the length 
of the longest train. 

When a curve occurs within the first 20 feet of rise of 
a grade. 

When curvature is in no sense limiting. 

(b) Compensate 0.035 per degree: 

When curves are between one half and three quarters 
as long as the longest train. 

When the curve occurs between 20 and 40 feet of 
rise from the bottom of the grade. 

(c) Compensate 0.04 per degree: 

Where the curve is habitually operated at low speed. 

Where the length of the curve is longer than three 
quarters of the length of the longest train. 

Where elevation is excessive for freight trains. 

At all places where curvature is likely to be limiting. 

(d) Compensate 0.05 per degree wherever the loss of eleva- 
tion can be spared. 

64. Maximum Trains. It is evident that grades and 
curves, by their resistances, fix a limit to the weight of a 
train which a given engine can haul over them. 

To find an expression for the maximum train which a given 
locamotive can haul over a given compensated grade: 

Let P = the draw-bar pull of the locomotive in pounds; 

T’ = weight of paying load in tons per maximum train; 
W’ = weight in tons of cars carrying the load 7”. 


Then for uniform motion at a given velocity, 


sh Wee a) = 2 (6) 


fy Ys FIELD ENGINEERING 


Let ¢ = the average load of one car in tons, and w = the 
average weight of one car including load in tons. The num- 
ber of cars, n, all assumed loaded to the average, will then 

Ty 
be ey 


/ 
Then W’ + T’ = nw = eee and substituting in eq. (6) 


ras le 
ums”: (, Sy 7) 7) 

In this equation g = the resistance per ton due to uniform 
motion, g’ = the resistance per ton due to the maximum 
grade opposed to the direction of the train. 

For accelerated motion the reaction of inertia of the train 
must be added to the above resistances. This is estimated 
at 3q, in order that a train starting from the rest may acquire 
the requisite maximum velocity, even on a maximum grade, 
in a reasonable time, say from three to six minutes. There- 
fore, for accelerated motion, 


we derive 

















t P 
sks =a ee) 3 | 
aCe ® 
The value of q in eq. (2) may be substituted in eq. (8) 
and we obtain - 
sb ipbe ite i . 
n(3 (Ki + 30.6 — oats) 3 ; 9) 
2 \" 4.08 + 0.152w 
Also, for the weight of maximum train and load 
W+T'= f Said Se Sch usar (10) 
- (3 (V + 39.6,— OB) : 
2\° 4.08 + 0.152w : 


which is the expression required. 
When there is no grade, q’ becomes zero, hence for a level 
track eqs. (8) and (9) become 


(ie eae (8a) 


T’ Fy | (9a) 





w 3 (" 333006) = ste) 
2 


2\ 4,08 + 0.152w _ 


wy 


MAXIMUM ECONOMY IN GRADES, ETC. 30 


where 7’, = the weight of paying load in tons per maximum 
train on a level grade. 
65. An engine-stage is a division of the road to which 
a locomotive is limited, and over which it regularly hauls 
a train. Its length varies, on existing roads, from 50 to 
200 miles or more, depending on the grades, on the length 
of the whole line, and on the distance between points favor- 
able for the location of shops, etc. The average engine- 
stage on American roads is not far from 100 miles. If there 
are to be several engine-stages on the proposed line, the prob- 
lem of maximum economy of grade must be solved with 
reference to each of them separately. 
Let L = length of engine-stage in miles; 
e = that part of expense per train mile which varies 
with grade (in dollars); 
A = average annual paying freight in tons moving in 
‘one direction, and 
a = average annual paying freight in tons, moving in 
the opposite direction; and if these are not equal, let A‘ be 
greater than a. Now 17”, eq. (9), is the maximum train-load 
which, at a velocity V, should be hauled up steepest grade 2’, 


opposed to the direction of the tonnage A; hence S = the 
number of trains per annum; and since each train must go 
ae = the total train-mileage per annum. 

If there were no return tonnage, the annual expense charge- 
able to A would be 2a but since some of the cars return 
loaded with the freight a, these are not chargeable to A, and 
must be deducted from the above expression. Hence if we 
denote the annual expense of engine-mileage by 2, 

2A — a)Le 
in which the value of the maximum grade 2’ is involved in 
the value of 7”. 

But we may obtain an expression for x in terms of 2’; 
for, at any given velocity, the resistance, go, on a level is 
equal to the resistance due to a certain grade 2, the value 
of which is, by eq. (3), for uniform motion, 


‘ zy = 2.64q0 


and return, -. 


34 FIELD ENGINEERING 


So the resistance, g, to motion up a grade 2’ is equal to the 
resistance due to some grade z = 2.64q, the total resistance 
being that due to the combined grades z + 2’. Now, since 
‘the gross weight of a maximum train, under a constant 
engine power, is inversely as the resistances, we have, for 
conditions of accelerated motion: 


al 2 al! Pe 5 : 2 + 2’ 

whence se 
y tes 0 20 
Lint z+ 22! 
in which 7)’ =maximum train load on a level line. Sub- 
stituting this value of 7’ in eq. (11) we have 
Ys 
ae an a omabayny (13) 

which is the complete expression for « = f(z’) required. 

66. Graphical Solution. Could we also find a com- 
plete expression for y = f’(2’), we might then proceed to 
find, by analysis, that value of 2’ which would render x + y = 
a minimum. But the value of y cannot be formulated, since 
it depends on the accidental features of the country through 
which the line passes; it can only be determined for any 
given value of 2’ by an estimate based on the survey. We 
therefore resort to a graphical solution. 

Eq. (13) is the equation 
of a curve in the plane ZX, 
Vig. 2. If we assume sey- 
eral values of z’, and cal- 
culate the corresponding 
values of x, we may lay 
these off by scale on the 
axes of Z and X respec- 
tively, and so obtain sev- 
eral points through which 
the curve of annual expense 
may be drawn. We then 
make estimates of the cost 
of constructing the road 
at the same values of 2’, and taking the annual interest of 
each estimate as an ordinate y to OZ in the plane ZY, we lay 





(12) 








MAXIMUM ECONOMY IN GRADES, ETC. 35 


it off to scale at the proper height, thus obtaining a series of 
points in the plane ZY, through which the curve of annual 
interest on first cost may be drawn. If now we suppose the 
plane ZY to be revolved to the left about the axis OZ until 
it coincides with the plane OX, as in Fig. 2, we shall see 
that the two curves are convex to OZ and to each other. 
The shortest horizontal line intercepted by them indicates 
the minimum value of («+ y), and the point where this 
line cuts the axis OZ indicates the corresponding value of 
2’, which is*the one required. If tangents be drawn to the 
curves at the points where the shortest horizontal line inter- 
sects them, the tangents will be parallel to each other. Any 
convenient scales may be used to lay off the values of 2’ 
and zx, provided that the values of z and y be laid off to the 
same scale. It is well to reduce all the values of x by an 
~amount common to them all, and the same with respect to 
values of y, before laying them off to scale. This will bring 
the two curves nearer together without altering their form. 

67. To facilitate the calculation of x, a table might be 
computed giving values of 7 for several locomotives and 
based on eq. (9). 

The value of x can be found by multiplying the reciprocal 
of T’ by (2A — a)Le. 

The data for a single locomotive and train will be taken as 
_ follows: E 


t = 30 tons of freight per car load; 
w = 45 tons per car and load; 
V. = 12 miles; 
P = 20,000 pounds. 


Substituting these values in eq. (9) and assuming differ- 
ent rates of grade, we can find the maximum loads of freight 
which the locomotive can haul up these grades. Further 
data and a complete solution of the problem are given in 
§ 69. . . 

68. In the given solution, the grades have been compen- 
sated. Grades less than the ruling grade need not neces- 
sarily be reduced for the curves upon them, unless the sum 
of the grade and the curve-equivalent exceeds the ruling 
grade. 


| 


36 FIELD ENGINEERING 


69. For an example, let us suppose that a certain engine- 
stage is to be 100 miles long, and that an estimate of the cost 
of construction has been made, based on a ruling grade of 
52.8 feet per mile against the heavier traffic, and that the 
annual interest on the estimate amounts to $168,000. 

Let us further suppose that the average traffic in one direc- 
tion is estimated at 375,000 tons per annum, and in the other 
direction at 125,000 tons, and that the part of the train- 
mile expense due to grade is 80 cents; hence (2A — a)Le = 
$50,000,000. We are now required to find the most eco- 
nomical ruling grade. 

First, select at least two other possible ruling grades, and 
having made an estimate of the cost of constructing the 
road upon each, take the annual interest of each, as in the 
first case. 

Let us suppose the two ruling grades thus selected to be 
73.92 and 31.68 feet per mile, or 1.4 feet per station and 
0.6 feet per station, and the interest on the estimates to be 
$145,596 and $204,388 respectively. Then the informa- 
tion may be grouped as in the table below. 























1 . 
G y Tp x | aty Zz 
iL 4 ba 5096s SPY hae oe nes Selene 73.92 
123 149 886 
Vee 155 050 .0023170 115 850 270 900 63.36 
is 161 088 .0021678 108385 | 269473 58.08 
1.0 168 000 0020175 100 875 268 875 52.80 
059) 2175786 .0018677 93 385 268 171 47.52 
0.8 | 184 446 .0017180 85 900 270 346 42.24 
0.7 | 193980 
0.6 204 888) er dlte eeceeary We te ee 31.68 





The values of y in the second column are obtained by inter- 
polating by second differences from the interest on costs of 
construction of the 1.4, 1.0, and 0.6 per cent grades. TJ” is 
obtained from eq. (9), using the data in §67. The fourth 


1 
column is obtained by multiplying 7 by (2A — a)Le, the 


numerical value of the latter being $50,000,000. We observe 
that the values of x and y increase in opposite directions, 
and that the minimum yalue of (x + y) will occur on a grade 
of about 0.9 per station, therefore this will be the most 
economical grade for the given conditions. As previously 


MAXIMUM ECONOMY IN GRADES, ETC. 37 


noted, the value of (2 + y) will not be the annual outlay 
of the road, or engine stage, because many items of expense 
which are independent of the ruling grade are not included 
in the value of 80 cents per train-mile. 

As a general proposition it is advisable to increase the size 
of locomotives. Certain limitations to increase of size 
occur, however, such as strength of bridges, the amount of 
traffic available, and so on. 

The problem may be pursued further with other locomo- 
tives and loadings, in which case it is to be remembered that 
as the size of the locomotive increases, the value of_e will be 
changed. Also the velocity may be increased or decreased, 
and if this is done the value of the draw-bar pull will change. 

70. Pusher Grades. Since z, eq. (11), varies directly 
as L, it is important that an engine-stage having heavy 
-grades should be short. Its length, however, must be 
consistent with the economical length of the adjoining 
engine-stages, and with the amount of work which a loco- 
motive ought to perform daily. The most favorable condi- 
tion for a road would be that in which all the engine-stages 
were operated at equal expense. But if, to secure this re- 
sult, the engine-stage of heavy grades must be unreasonably 
reduced in length, it will be better to adapt the grades to the 
use of two locomotives per train. 

71. Maximum Return Grades. The maximum grade 
z’, opposed to the heavier tonnage A, having been determined, 
we have now to consider what is the limit to grades in the 
opposite direction. The locomotives are supposed to haul 
their maximum loads in moving the tonnage A, and since 
the return tonnage, a, is less than A, the locomotives, in 
returning, will not be worked to their full capacity if they 
encounter no grades steeper than z’. We therefore have a 
margin of power in the returning locomotives which may 
be taken advantage of to cheapen the cost of construction, 
or to shorten the line, by introducing grades, steeper than 
z’, against the lighter traffic. 

The weight of a maximum train moving up the grade 2’ 
is, eq. (10), W’ +7"; the wefght of the train returning 
will be . 

aaa dan wireatie, ae j 
tpl ar +9) 


4 


38 FIELD ENGINEERING 


Substituting this in place of (W’ + 7”), eq. (10), and solving 
for q’, we find the resistance due to a maximum grade opposed 
to the returning train. Whence, by eq. (3), if we let Z = 
the maximum return grade, ; 


Pe 

WT saad. ; 
( gacik a} q | 
Inasmuch as the value of Z varies with every change made 
in 2’, the engineer, when estimating ‘the cost of construction 
upon the basis of any maximum grade of 2’, should take care 
that the return grade of Z nowhere exceeds its limits as 
given by the last eq. (14).. In the example, § 69, 2’ = 52.8 
approx.; hence 7’ = 495.6, eq. (9). Substituting these 
values in eq. (14), we find that Z = 127.8, which is there- 
fore the limit for return grades in this case. These grades 
should of course, be compensated. 

72. Undulating Grades. Suppose that 30 miles per 
hour is a safe maximum velocity for a freight train going 
down any grade. Assuming that a train starts down a 
grade at this velocity, we may find the value of the grade 
which will cause the train to continue at a constant velocity, 
without the use of steam or brakes. 

Designate the value of Gm in eq. (3) by z, and we have 


z = 2.64¢q 





Z= gail (14) 


‘Substitute the value of g from eq. (2) in this and we get 








rae Vieouse.e gosh) 
ei 2.64 ( 4.08 - 0.152w (15) 
When V = 80 this becomes 
os 30 + 39.6 — 0.031w 
re 2.64 ( 4.08 + 0.152w ) (15a) 


As a speed of 30 miles an hour is considered unobjection- 
able, the grade z which induces it cannot be so, provided, 
of course, it does not exceed the values of 2’ or Z which 
were found as the most economical. For the extra work 
done by the engine in ascending one grade z is utilized in 
descending the next; and the net result is the same as though 
the two were replaced by a uniform grade. The engineer 


MAXIMUM ECONOMY IN GRADES, ETC. 39 


therefore is not warranted by economic considerations in 
reducing undulating grades which do not exceed z to a uni- 
form grade, when to do this would cause any increase in the 
cost of construction, unless z exceeds the grades 2’ or Z of 
maximum economy. 

73. But when grades exceed z, eq. (15a), the resulting 
speeds of the maximum train become too great, and the 
necessary application of the brakes absorbs a portion of the 
power previously expended in gaining the summit, which 
is thus worse than wasted, since it increases the wear and 
tear of machinery and track. Therefore the engineer is 
justified in spending a certain sum of money in reducing 
grades which exceed z to that limit. A calculation of the 
loss of power due to the use of brakes on a grade, and of the 
cost of that lost power, together with the resulting wear 
and tear per annum, will give the interest on the sum that 
may be justifiably spent in reducing the grade from its 
position of cheapest construction. 

74. The limit z is not constant, but depends on the weight 
of the maximum train, which in turn depends on 2’. It will 
not be the same in both directions unless A = a, giving 
z’ = Z. In the example §69, W’-+ 7” = 743.4; hence, 
eq. (15a), 2 = 16.24 descending in the direction of the traffic 


A. Also W’ + a T’ = 413.0, whence z = 23.04 descending 


in the opposite direction. These are the limits in this case 
at which undulating grades cease to be profitable. 

75. Comparison of Routes. We have finally to con- 
sider the method for selecting the best line from several 
proposed routes. For this purpose we determine the most 
economical grade on each route thought worthy of con- 
sideration, and calculate the interest on the entire cost of 
constructing the line with that ruling grade, and also the 
annual expense of operating the line, and take the sum 
of the two. That route is best in respect to which this 
sum is the least. 

76. The value of saving one mile in distance on any route 
is found by dividing the sum of the annual operating expense 
and the interest on the cost of construction by the rate of 
interest, and the quotient by the length of the line in miles. 

77. Conclusions. We have now fully discussed the 


40 FIELD ENGINEERING 


theory and developed the formulas necessary to the deter- 
mination of the most economical grades; but the value of 
the results in a given case depends upon the correctness of 
the engineer’s estimates which enter into the formulas. These 
may seldom prove precisely accurate, yet, if he can bring 
them within definite limits, he may determine the grades 
of maximum economy within corresponding limits. But in 
the case of a railroad already finished and in full operation, 
where the elements of first cost, of the existing traffic and 
of the operating expenses are fully known, an investigation 
by means of the foregoing formulas furnishes a critical test 
as to the economy of any proposed alteration of the line. 
In several instances which might be named the trunk line 
roads have been justified in incurring very heavy expense 
in reducing distances and grades for the cheaper handling 
of greatly increased traffic. 


CHAPTER IV 
LOCATION 


78. A railroad is said to be located when its center line is 
established on the ground in the position which it is intended 
finally to occupy. The location is made by an engineer corps 
similar in its organization to that employed on preliminary 
surveys. A fully equipped engineer’s transit is required. 
The other instruments are the same as before except that 
the target rod may replace the self-reading rod. The mag- 
. netic needle is never used upon the center line, except as a 
rough check on the transit work. It is used, however, to 
obtain the direction of property lines, roads, and other 
topographical data. 

79. The remarks upon transit work in the preceding . 
chapter apply to the running of straight lines on location. 
All field-work on location should be done with accuracy and 
fidelity. No guesswork, nor rude approximations, are to be 
tolerated. All transit points are made as secure and per- 
manent as possible, and the more important ones are guarded 
by other transit points set in safe positions near by, their 
distances and directions from the main point being recorded. 

The stakes for the stations are made neatly, and somewhat 
uniform in size, and they are firmly driven. Sometimes a 
small plug is driven down flush with the surface of the ground 
to indicate the station point, and the stake is then set near by 
as a witness. 

In locating a very long tangent the greatest care is re- 
quired to make it straight. If the tangent is produced from 
point to point by backsights and foresights, the observation 
should be repeated in every instance with reversed instrument, 
to eliminate any possible lack of adjustment, and to check 
any accidental error. (Indeed it is proper to observe this rule 
on curves, as well as on tangents.) When some object in the 
horizon can be used as a foresight, it is preferable to set the 


, Ad 


| 


42 FIELD ENGINEERING 


instrument by this rather than by a backsight. For final 
location, the line should be cleared to give as continuous a 
line of sight as possible, but in case of an obstacle which 
cannot be removed at the time, at least two independent 
methods of passing it should be employed, so that there may 
be a check upon the alinement beyond. 

80. The leveler selects his benches far enough from the 
line to prevent their being disturbed during the construction 
of the road. They should be nearly at grade, as a rule, 
though it is well to leave a bench near.a water-course for 
reference in laying out masonry or trestle-work. The rod- 
man holds the rod at every station, and at every point on the 
center line where the slope changes direction, so that these 
points may be accurately defined on the profile. When he 
uses a target rod, he sets the target as directed by the leveler 
and after clamping it, takes the reading. He reads to thou- 
sandths upon turning points and benches, but only to tenths 
of a foot elsewhere, and announces the readings to the leveler 
for record. He also records the readings upon turning points 
and benches in his own book asa check. At the close of each 
day the leveler and rodman compare notes, and draw a pro- 
file of the line surveyed. (See also §$§ 28, 29, 30.) 

81. The fixing of the grade-lines upon the profile is 
one of the most important operations connected with the 
location. Itis usually performed by the engineer in charge of 
the locating party, as being most conversant with the general 
character and detailed requirements of the line. The maxi- 
mum gradients will have generally been determined in advance 
from the preliminary data by the principles laid down in the 
preceding chapter, but the position of each grade-line relative 
to the profile of the surface must be left to the judgment and 
skill of the engineer. In general, the grade-line is so placed 
as to equalize the amounts of excavation and embankment, 
but there are various exceptions to this rule. Thus, the exca- 
vation may be in excess: first, when it is necessary to pass 
under some other road or highway, the grade of which cannot 
be changed; second, when valuable property is to be avoided, 
the appropriation of which would cost more than the excava- 
tion; third, when the grade is at the maximum near a sum- 
mit, and cannot be raised parallel to itself without incurring 
too great an expense for masonry, etc., at some other part of 


LOCATION 43 


the line. The embankment may be in excess, first, when the 
country is flat and wet, in order to keep the road-bcd wall 
drained (the grade-line should be at least two feet above the 
average level of the surface, or above the high-water mark, if 
the district is subject to overflow); second, in approaching a 
stream, where it is necessary to raise the grade above the 
requirements of navigation; third, when the cuttings on the 
line are largely in solid rock, and a cheaper material for 
embankments may be conveniently had at other points; 
fourth, in a, district subject to heavy drifts of snow by which 
deep cuts would be lable to be obstructed; jifth, in side-hill 
work, where there is danger of land-slips; siath, when it is 
determined to supply the place of a portion of an embankment. 
by a timber trestle-work or other viaduct. 

The apparent equality of cut and fill on the profile does not 
. represent an equality in fact, owing to the different bases and 
slopes of the sections adopted, and to the various inclinations 
of the natural surface transversely to the line. This is espe- 
cially true in side-hill work, where there are both cut and fill 
at every point, while the profile shows very little of either. 
In the latter case it is an excellent plan to combine with the 
profile of the center line the profiles of parallel lines ten 
or twenty feet either side of the center, and drawn with differ- 
ent colored inks, as these will indicate tolerably well the 
relative amount of cut and fill required. But after the grade 
has been thus chosen, the only safe method in side-hill work 


~ is actually to compute the amounts of excavation and embank- 


ment from cross-sections, mark the amount for each cut and 
fill on the profile, and compare the results. Any changes 
required in the grade or alinement may then be discovered 
and effected before the work of construction has begun. 


CHAPTER V 


SIMPLE CURVES 


A. Elementary Relations 


82. The center line of a located road is composed alter- 
nately of straight lines and curves. 

The straight lines are called tangents because they are 
laid exactly tangent to the curves. A tangent may be in- 
definitely long, but should never, as a rule, be shorter than 200 
feet between two curves which deflect in opposite directions, 
nor shorter than 500 feet between curves which deflect in the 
same direction. A curve should not be less than 200 feet 
long. When a tangent is said to be straight, the meaning 
simply is that it has no deflections to the right or left; for 
since it follows the surface of the ground, it evidently has as 
many undulations as the ground. But if we conceive a ver- 
tical plane to be passed through the line, a horizontal trace of 
this plane will accurately represent the line; and so, if we con- 
ceive a vertical cylinder to be passed through a curve on the 
surface of the ground, a horizontal trace of that cylinder will 
accurately represent the curve, since all distances and angles 
are measured horizontally, whatever be the irregularities of 
the surface. In all problems, therefore, relating to this sub- 
ject, we may consider the ground to be an absolutely level 
plain. 

83. A Simple curve is a circular are joining two tan- 
gents. It is always considered as limited by the two tangent 
points, and any part of it beyond these points is called the 
curve produced. The first tangent point, or the point where 
the curve begins, is called the Point of Curve, and is indicated 
by the initials P.C. The point where the curve ends, and the 
next tangent begins, is called the Point of Tangent, and is 
indicated by the initials P.T. When accessible, these points 
are always occupied by the transit in the course of the survey, 

44 


SIMPLE CURVES 45 


and the plug driven to fix the point is guarded not only by 
the usual stake bearing the number of the station, but also by 
another bearing the proper initials, the ‘‘ degree” of the 
curve, and an “R”’ or “L”’ to indicate whether the deflection 
is to the Right or Left. 

84. A simple curve is designated either by the radius, R, 
er the degree of curve, D. . 

The Degree of Curve, D, is an angle at the center, sub- 
tendea by a chord of 100 feet. It is expressed by the number 
of degrees and minutes in that angle, or in the arc of the 
curve limited by the chord of 100 feet. Therefore D equals 
the number of degrees of arc per station. 

The radius R and degree of curve D can be expressed in 
terms of each other. 

Let ab, Fig. 3, be a chord 

‘of 100 feet subtending an 

are described with a radius 
aO=RK from the center O. Q 
Then, by definition the angle 
bOa=D.  Bisect the angle q 
bOa by a line Og, and this line 

will also bisect the chord ab Braig 

and be perpendicular to it; 

and in the right-angled triangle bgO we have 


bg = Ob X sin bOg 
= . 
100 = Rsin 3D 


Hence, to find Radius in terms of Degree of Curve: 


nije hoo 
ie sin 4 Ga 
and to find Degree of Curve in terms of Radius: 


Skee OU) . 
sin 3D = R (17) 
It is the practice of English engineers to assume the radius 
at some round number of feet and calculate the degree of 
curve, which is therefore fractional. In America, on the con- 
trary, the degree of curve is assumed at some integral number 
of degrees or minutes, and the radius deduced from this. 


46 FIELD ENGINEERING 


Example.—What is the radius of a 3° 20’ curve? 


50 log 1.698970 
$D = 1° 40’ log sin 8.463665 


Ans. RB = 1719.12 log 3.235305 


Thus the second and third columns of Table I have been 
calculated, giving at once # or D in terms of the other. 
Ezxample—What is the degree of curve when the radius is 
600 feet? 
50 log 1.698970 
R=600 log 2.778151 


1D = 4° 46’ 48.73 log sin 8.920819 
Ans. _D = 9° 33’ 37.46 


Measurement of Curves. 


85. A railroad curve is always asumed to be measured with 
a 100-foot tape, and as the tape is stretched straight between 
stations it cannot coincide with the arc of the curve, but 
forms a chord to the arc, as in Fig. 3. Consequently the 
curve as measured from one tangent point to the other is an 
inscribed polygon of equal sides, each side being 100 feet. 
The sum of these sides (with any fraction of a side at either 
end of the curve) is called the Length of curve, L.. This length 
L is evidently a little less than the length of the actual arc 
between the same points, but the latter we very seidom have 
occasion to consider. 

86. If the chain lengths were taken on the arc instead of as 
chords of the curve, the degree of curve would be inversely 
proportional to the radius, and since the are whose length is 
equal to radius contains 57.3 degrees nearly, we should have 


PDP Sto LOU ote 
or 
Bs 5730 


Soe PT) 





a convenient formula, but only approximately true when D 
is small, and seriously at fault when D is large; the error in- 
volved being proportional to the difference in length of a 
100-foot chord, and the are which it subtends, 


SIMPLE CURVES 47 


87. The Central Angle of a simple curve is the angle | 
at the center included between the radii which pass 
through the tangent points (P.C.) and (P.T.). It is there- 
fore equal to the number of degrees contained in the entire 
are of the curve between these points. The central angle 
will be designated by the Greek letter A (delta). 

From the definitions of the length and degree of curve we 
have the proportion, . 
Deo aLUU wel 
Hence, to find the Length of curve in terms of the central 
angle: 


A 
L = 1005 (18) 


_ Example.—What is the length of a 4° curve when the cen-. 
‘tral angle is 29°? 


D= 4° gud A=29° { 4)2900 
Ans. L=7stations+25 feet | 725 feet 





Note.-—When A or D are not in exact decimals of a degree, 
reduce both values to minutes before dividing one by the 
other. 

Example.—What is the length of a 4° 20’ curve, when the 
central angle is 29° 35’? 


D, = 260’ and A = 1775’ es 
Ans. L =6 stations + 82.69 feet 682.69 feet 





To find the Central angle in terms of the length and degree 
of curve: 
A = =—— (19) 


Ezxample.—What is the central angle of a 5° curve 730 feet 


long? 
5 X 730 


F 21003 20 36 a5) 


UD sae L =,730, 
Ans. A = 36° 30’ 


To find the Degree of curve in terms of the length and 
— central angle: 


(20) 


48 FIELD ENGINEERING 


Example.—What is the degree of a curve 8 stations long, 
and having a central angle of 26° 40’? 


L = 800, A = 1600’, 100 sated = 200’ 


800 
Ans. D = 3° 20’ 


88. If two tangents, joined by a simple curve, are prox 
duced (one forward and the other backward) until’ they 
intersect, the point of intersection, V (Fig. 4), is called the 
vertex, and the exterior or deflection angle which they make 
with each other is equal to the central angle, A. 





Fig. 4. 


The Tangent-distance, T, is the distance from the ver- 
tex to either tangent point; thus in Fig. 4, 7 = AV = VB. 

The Long Chord, C, is the line AB joining the two 
tangent points. 

The Middle-ordinate, M, is the line GH, joining the 
middle point of the long chord with the middle point of the 
curve. - 
The External distance, ZL, is the line HV, joining the 
middle point of the curve with the vertex. 

We observe that both the middle-ordinate, M, and the 
external distance, Z, are on the radial line joining the center, 
O, with the vertex, V, and that this line is perpendicular to 
the long chord, C, also, that it bisects the central angle 
AOB =A, and its supplement AVB. (Table XLI, 14.) 
We also observe that the angle VAB = VBA = 3A (Table 


SIMPLE CURVES AQ 


XLI, 20); and if in the figure we draw the two chords AH 
and BH, the angle BAH equals one half the angle BOH, or 
BAH = ABH = {A (Table XLI, 18); also the angle VAH = 
VBH = 1 

89. If we have laid out two tangents on the ground, inter- 
secting at V, and have measured the angle, A, between them, 
we may then assume any other one of the elements of a simple 
curve before mentioned, and calculate the rest. If we 
assume D, for instance, we then find & by eq. (16) or by 
Table I. 

Then having A and R, we may proceed to calculate the 
other elements as they are needed. 

90. To find the Tangent-distance in terms of the Radius 
and Central Angle: 

In the right-angled triangle VOA, Fig. 4, we have 


VA = OA X tan VOA 
T = R tan 3A (21) 


Otherwise, approximately: In Table III, opposite the 
central angle, take the value of T for a 1° curve and divide 
it by the degree of curve D. If desirable, add the correction 
taken from Table II, corresponding to D. 

Example.—What is the tangent distance of a 4° curve with 
a central angle of 30°? 





D=4° R (Table I) log 3.156151 
A= 305 A> Bat 15" log tan 9.428052 
Ans. T = 383.89 feet log 2.584203 
Otherwise: 
By Table III, 4)1535.3 
Approximate ans. 383.82 
Correction from Table II .08 
Ans. T = 383.90 feet 


91. To find the Long Chord C, in terms of Radius and 
Central Angle: 
In the right-afigled triangle BOG, Fig. 4, we have 


50 FIELD ENGINEERING 


BG = BO X sin BOG 
or 
3C = Rsin 3A 
Ans. C =2 sin $A (22) 


But in case A can be divided by D without a remainder, 
that is, if the curve contains an exact number of stations 
(not exceeding 12), we may take the long chord at once from 
Table IV. 

Example.—What is the long chord of a 3° 20’ curve with 
a central angle of 36° 40’ ? 


2 | log 0.301030 
Dea 3” 20 Ahr apie 1) log 3.235305 
A = 36° 40’, $4 = 18° 20’ log sin 9.497682 


Ans. C = 1081.48 feet log 3.034017 


Otherwise: 





20 
= = = a3 = 11 stations 


And by Table IV, C = 1081.48. 

92. To find the Middle-ordinate M, in terms of Radius 
and Central Angle: 

From Fig. 4 


M =GH = OH — OG = Rf — Ff cos 3A 





or 
M = R(1 — cos 3A) 


and since the versine = 1 — cosine 
M = R vers ZA (23) 


But in case A can be divided by D without a remainder, 
that is, if the curve contains an exact number of stations 
(not exceeding 12), we may take the middle-ordinate at 
once from Table VI. 

Example.—What is the middle-ordinate of a 4° 30’ curve 
with a central angle of 40° 30’? 


D 4° 30’, R (Table I) log 3.105022 
A = 40° 30’, 3A = 20° 15’ log vers 8.791049 


Ans. M = 78.717 feet 1.896071 


SIMPLE CURVES OL 


Otherwise: 
Roe} 
D® i455 
and by Table VI, M = 78.717. 
For additional formulas for the middle-ordinate, see 
Chapter XVII, Track Laying.’ @ 
93. To find the External Distance E in terms of Radius 
and Ceniral Angle, 
From Fig, 4, 


E = VH = VO — HO 


R 1 
B= R= R( -1) 


and since the ex sec = secant — 1 we have, 


= 9 stations 


or 








E = R ex'sec, 34 (24) 


Otherwise, approximately: 

In Table III, opposite the central angle, take the value of 
E for a 1° curve, and divide it by the degree of curve D. 
If desirable, add the proper correction corresponding to D, 
taken from Table II. 

Example.—What is the external distance EF of a 7° 30’ 
curve when the central angle is 60°? 








Dims (130, A (lable Ty log 2.883371 
A = 60°, 1A = 30° log ex sec 9.189492 
Ans. H = 118.27 feet log 2.072863 
Otherwise’ 
By Table IIL 7.5)886.38 
Approximate ans. 118.184 
Correction for V = 7° 30’ (Table IT) 084 
Ans. H = 118 .268 


94. But, instead of assuming D or R, we may prefer, or 
may find it necessary to assume, some other element of the 
curve, the central angle being given. 

If we assume the tangent distance, then: 

95. To find the Radius and Degree of Curve in terms 
of the Tangent-déstance and Central Angle. 


52 FIELD ENGINEERING 


From eq. (21), and by Table XLII, 40, we have 
R = T cot.4A (25) 
Otherwise, approximately: 


Divide the tangent of a 1° curve found opposite the value 
of A in Table III, by the assumed tangent-distance; the 
quotient will be the degree of curve in degrees and decimals. 

Example.—The exterior angle at the vertex is 54°, and the 
tangent-distance must be about 700 feet. What shall be the 
degree of curve? 


A =54°,. 3A = 27° log cot 0.292834 





T=700 2.845098 
log R = 3.137932 
Ans. By Table I D =4° 10’ + 
Otherwise: 
By Table ITI, 700)2919.4 
Ans. D = 4°10' 15” 4°1706 


But as it is difficult to lay out a curve when D is fractional, 
we discard the fraction and assume 4° 10’ as the value of D. 
This may require us to recalculate the value of 7, which we 
do by eq. (21) and find 7 = 700.8 feet log 2.845596. If the 
other elements are required, they may be calculated by 
eqs. (22), (23), (24), or directly from T and A, as follows: 

96. To find the External distance £, in terms of the 
-Tangent-distance and Central Angle. 

In Fig. 5 we have given 
AOB =A and AV = 7, to find 
HV = E. In the diagram draw 
the chord AH, and through H 
draw a tangent line to intersect 
OA produced in J, and join VI. 

Then HI is parallel to BA, 
and since HI = AV = T, and 
AI = HY = E, VI is parallel to 
HA, and VIH = HAB = ja. 
(Table XLI, 18.) 

In the right-angled triangle VHI we have 





E1Gs25. 


SIMPLE CURVES 53’ 


AY = AT Xitan Vi 
or E =T tan iA (26) 


Example.—The angle at the vertex being 54° and the tan- 
gent-distance 700.80 feet, how far will the curve pass from 
the vertex? 


T =700.80 (from last example) 2.845596 
A = 54°, 7A = 138° 30’ log tan 9.380354 


Ans. °° E = 168.25 feet ; log 2.225950 


(For the formulas by which to find the long chord and 
middle-ordinate in terms of the tangent-distance and central 
angle, see Table XLII, 12 and 13.) 

97. Again, it may be necessary to assume the external dis- 
_tance in order to determine the proper degree of curve. 

To find the Radius and Degree of Curve in terms of 
the External distance and Central Angle: 

By eq. (24) 

page ee (27) 
ex sec 3A 
Otherwise: : 

In Table III divide the external distance of a 1° curve, 
opposite the given value of A, by the assumed external dis- 
tance; the quotient is the degree of curve required. 

Example.—The angle at the vertex being 24° 30’, the curve 
- is desired to pass at about 65 feet from the vertex. What 
is the proper degree of curve? 


if ==,00 log 1.812913 
A = 24°30’, - 4A = 12°15’ log ex sec 8.367345 





log R = 3.445568 
Ans. By TableI D = 2° 03’ + 
Otherwise: . 
_ By Table III 65)133.50 
Ape D2 030 14"" 2°.0538 


We may therefore assume a 2° curve, unless required by 
the circumstances to be more exact, when we might use a 
2° 03’ curve. Assuming a 2° curve, we have by eq. (24) 


E = 66.75 log 1.824460 


54 FIELD ENGINEERING 


Having decided on the degree of curve, we may calculate 
the remaining elements by eqs. (21), (22), (23), which is 
always the better way, but we may calculate them directly 
from F and A. 

98. To find the Tangent-distance in terms of the 
External distance and Central Angle: 

From eq. (26), and by Table XLII, 40, 


T = EF cot tA (28) 


Example.—The angle at the vertex is 24° 30’, and the 
curve passes 66.75 feet from the vertex. How far are the 
tangent points from the vertex? 


E = 66.75 (from last example) log 1.824460 
A = 24°30’, iA = 6°07’ 30” log cot 0.969358 


Ans. T = 622.04 feet 2.793818 


99. Remark.—Eqs. (27) and (28) are particularly useful 
in defining the curve of a railroad track where all original 
points are lost. Produce the center lines of the tangents 
of the curve to an intersection V, and there measure the 
angle A. Bisect its supplement AVB, and measure the dis- 
tance on the bisecting line from V to the center line of the 
track. This will give VH = HE. Then R and T may be 
calculated, and the distance 7’ laid off from V on the tan- 
gents, giving the tangent points A and B. 

(For the formulas by which to find the long chord and 
middle-ordinate in terms of # and A, see Table XLIII, 26 
and 27.) 

100. Again, having only ane central angle given, we 
may assume the long chord, or the middle-ordinate, and from 
either of these and the central angle calculate the remaining 
elements. Or, finally, the central angle being unknown, 
we may suppose any two of the linear elements given, and 
from these calculate the rest. As such problems have little 
practical value, their discussion is omitted. The requisite 
formulas for their solution are given in Table XLII, and the 
verification of them is suggested as a profitable exercise 
to the student. 


SIMPLE CURVES 55 


B. Location of Curves by Deflection Angles 


101. In order that the stakes at the extremities of the 
100-foot chords, by which the curve is measured, shall be set 
exactly on the are of the curve by transit observation, it is 
necessary at the point of curve, A, to deflect certain definite 
angles from the tangent AV. Let us suppose that in the 
curve AB, Fig. 6, the points A, a, b, c, d, etc., indicate the 
proper positions of the stakes 100 feet apart, and that OA 
is the radius of the curve. In the diagram join Oa, Ob, 
etc., and also Aa, ab, be, etc. Then, by definition, the angle 





Fig. 6. 


AOa = D, and by Geom. (Table XLI, 20 and 11) the angle 
VAa =4D. Therefore if we set the transit at A, and 
deflect from AV the angle 3D, we shall get the direction of 
the chord Aa, on which by measuring 100 feet from A we 
fix the stake, a, in its true position on the curve. So again, 
since the angle aOb, at the center, = D, the angle aAb, at 
the circumference, = 3D. If therefore, with the transit 
at A, we defiect the angle 3D from the chord Aa, we shall 
get the direction of the chord Ab; and when the stake b is 
on this chord it will also be on the curve, if b is 100 feet 
distant from a. Thus, in general, we may fix the position 
of any stake on the curve, by deflecting an angle $D from 
the preceding stake, and at the same time measuring a 
chain’s length from it,—the chain giving the distance, while 
the instrument at A gives the direction of the point. 

2D is called the deflection-angle of the curve; so that in 


56 FIELD ENGINEERING 


any curve, the deflection-angle is equal to one half the degree of 
curve. 

102. Since each additional station on the curve requires 
an additional deflection-angle, the proper deflection to be made 
at the tangent point from the tangent to any stake on the 
curve is equal to the deflection-angle of the curve multiplied 
by the number of stations in the curve up to that stake; 
or it is equal to one half the angle at the center subtended 
by the included are of the curve. 

103. It may happen that all the stations of a curve are 
not visible from the tangent point A. When this is the case 
a new transit-point must be prepared at some point on the 
curve, by driving a plug and center in the usual manner, 
and the transit moved up to it. Let us suppose that the 
point d, Fig. 6, has been selected for a transit-point, and 
that the transit has been set up over it. Before the curve 
can be run any farther, it is necessary to find the direction 
of a tangent to the curve at the point d. The curve be- 
yond d may be located by two different methods. 

First Method. Sight at station A with a vernier reading 
laid off equal to that used in locating the point d, but on the 
opposite side of zero. Turn the vernier to 0° and the tele- 
scope will then point along the auxiliary tangent (line dz). 
Plunge the telescope and locate the points e, f, etc., by de- 
flecting (4D), 2(D), and so on. This operation may be 
expressed by the following: 

Rule. To find the direction of the tangent to a curve 
at the extremity of a given chord, deflect from the chord an angle 
equal to one half the angle at the center subtended by the chord. 
‘(Table XLI, 20.) 

The tangent beyond the P.T. may be found by the above 
rule. 

104. Second Method. Compute a complete list of 
deflections as if the entire curve were to be located from the 
P.C. Locate the point d in the usual way. Set up at d, 
sight at A with a zero reading on the vernier. Turn the 
telescope outward until the reading which was used in 
establishing d is reached. The telescope will then be on the 
auxiliary tangent dz. Plunge the telescope and lay off the 
deflections for the points e, f, etc., as found in the original 
list. Any other transit station than A might have been 


SIMPLE CURVES 57 


sighted at provided the proper deflection were used. The 
general rule for this method is as follows: 

Rule for the Second Method of Deflections. When a set up 
is made at any point on a curve, an auxiliary tangent may be 
found at that point by sighting at any transit station of the curve 
with the deflection of the station sighted at laid off on the proper 
side of zero. 

Comparison of the Two Methods. The first method must 
always be used with compound curves, a set up being neces- 
sary at each P.C.C. With whole stations and whole degrees 
of curve neither method has any special advantage. The 
particular advantage of the second method is in the fact 
that a list of deflections can be computed and checked before 
any points on the curve are located, and this list may be 
used regardless of the number of set ups which are later 
found necessary. 

Caution. Extreme care should be taken that an tapos! 
combination of methods is not made. One method should 
be chosen and adhered to. 

-105. The center line is measured, and the trast num- 
bered regularly and continuously through tangents and 
curves from the starting point to the end of the work. It 
therefore frequently happens that a curve will neither begin 
nor end at an even station, but at some intermediate point, 
or plus distance. 

If the Point of Curve occurs a certain number of feet 
beyond a station, the first chord on the curve is composed 
of the remaining number of feet required to make 100. 

Any chord less than 100 feet is called a subchord. 

If a curve ends with a subchord, the remainder of the 
100 feet must be laid off on the tangent from the Point of 
Tangent to give the position of the next station, so that the 
stations may everywhere be 100 feet apart. 

106. The deflection to be made for a subchord is ae to 
one half the arc it subtends. 

Let c = length of any subchord in feet, 

d = angle at center subtended by subchord. 


Then, from eq. (22), by analogy 





c = 2R sin id (29) 
100 | 
But by eq. (16) as 3D 


58 FIELD ENGINEERING 





‘ Wh sin 3d 
oe c = 100 ain yD (30) 
ay sind = —~ sin 4D (31) 


00 


When D does not exceed 8° or 10°, we may assume without 
serious error that the angles are to each other as their sines, 
and the last two equations become 


(approx.) C= 100 4 (32) 
ety hae OL 
and id 100 (4D) (33) 


In curves sharper than 10° per station, the error involved 
in this assumption becomes apparent and must be corrected. 

107. If curves were measured on the actual are, then 
eqs. (82) and (33) would be true in all cases; but since a 
curve is measured by 100-foot chords, it is evident that if 
a 100-foot chord between any two stations were replaced 
by two or more subchords, these taken together would be 
longer than 100 feet, since they are not in the same straight 
line. Let us conceive the actual are of one station to be 
divided into 100 equal parts; since the arc is longer than 
the chord, each part will be slightly longer than 1 foot. 
Now if we take an are containing any number of these parts 
(less than 100), the nominal length of the corresponding 
subchord in feet will equal the number of parts, and the 
deflection for the subchord will be proportional to the num- 
ber of parts which the are contains. The deflection there- 
fore will be exactly given by eq. (83) if in that equation we 
let c equal the number of parts in the arc, or the nominal 
length of the subchord in feet. Having thus obtained the 
correct value of (3d), we may introduce it into eq. (29) or 
(30), and obtain the true value of the subchord, which will 
always be a little greater than its nominal value. 

Suppose, for instance, that the arc of one station is to be 
divided into four equal portions; then each subchord will 
be nominally 25 feet long; and by eq. (33) 


(2D) = 4(2D) (33a) 


SIMPLE CURVES 59 


which is the correct value of the deflection, whatever be the 
degree of curve. Substituting this value in eq. (29) or (80) 
we obtain the true value of the subchord, c, a little greater 
than 25; the ezxcess is called the correction of the nominal 
length. 

108. This correction for any given subchord bears an 
almost constant ratio to the excess of are per station, what- 
ever be the degree of curve. These ratios are shown in the 
following table for a series of subchords, and Table IV gives 
the length of actual arc per station for various degrees of 
curve. Subtracting 100 we have the excess of arc per sta- 
‘tion, and multiplying this excess by the ratio corresponding 
to the nominal length of subchord we obtain as a product 
the proper correction for the subchord. 


TABLE OF THE RATIOS OF CORRECTIONS OF SUBCHORDS 
TO THE EXCESS OF ARC PER STATION 








Nominal Nominal Nominal 
Length of Ratio. Length of Ratio. Length of | Ratio. 
Subchord. Subchord. Subchord. 
0 000 35 3807 70 356 
050 40 336 75 327 
10 9 45 858 80 287 
15 147 50 374 85 235 
20 192 55 883 90 169 
25 234 60 883 95 002 
30 273 65 874 100 000 











We observe that the largest correction is required by a 
subchord between 55 and 60 feet in length. 

Example.—lt is proposed to run a 14° curve with a 50-foot 
chain. What correction must be added to the chain? 


50 
se ° 2-$), 47.9 APS etalk Sas. © . 2°F -. 90 ! 
D.=44 1D=7 td i00 * @ 32.5 =93 2.00 
By eq. (30) in Rerbenh 40 
ie 00 eee ae 50,098 
sin 7 
Ans. Correction = .093. 

Or, by Table IV, length of are = 100.249 

excess of arc = . 249 

and by above table, ratio for 50 feet = 374 


Ans. Correction = product = .093 


8 


60 FIELD ENGINEERING 


Example.—The P.C. of an 18° curve is fixed at + 55 feet 
beyond a station. What are the nominal and true values of 
the first subchord, and what the proper defleetion? 


Nominal value = 100 — 55 = 45 feet 





Deflection =4d = a se 9° a4 U5 = 4 Ose 

and by eq. (30) ores 

True value =c = 100 eee = 45.148 

sin 9° 
Or, by Table IV, excess of arc = .412 
by above table, ratio for 45 feet = 358 
Correction = product = .147 
Ans. True value of subchord = 45.147 


Example.—The last deflection at the end of a 40° curve is 
found to be 6° 80’.. What are the nominal and true values 
of the last subchord? 

os id = 6° 30’, and by eq. eye 


6.5 





Nominal value, c = 100 20 = = 32.5 feet 
By eq. (80) is: ee 
True value, c = 100 Su ake = 33.098 feet 
sin 20 
Or by Table IV, excess of arc 40° = 2.060 
by above table, ratio for 32.5 feet =  .290 
Correction = product =  .597 


Nominal value of subchord = 32.5 





True value = 33.097 


109. The transitman keeps neat and systematic field- 
notes of all his operations with the transit, whether on tan- 
gents or curves. ‘The instrumental work is recorded on the 
left-hand page of the transit book so as to read upward on 
the page, while the right-hand page is reserved for sketches 
of prominent objects and for explanatory remarks. Thus 
the station numbers in the book advance in the same num- 
ber as the stations on the ground, and there can be no con- 
fusion as to the right and left of the line in making the 


SIMPLE CURVES 


61 


record. The complete topographic record is left for the 
topographer to make in a special book or on special paper. 


FORM OF TRANSIT BOOK 


Sta. | Point. | Plus. | Deflec. | Angle. 




















26 

25 

od yep, ee RS OR MY Cee Re a 

© PT +15 6° 18’ 

Te ee Ree EOS ee 6 

Soe aliesane entices tenes 4 

PIDdH ojstheesneD.: 2° 

20 ‘o) ~ 4° 48’ 

ROE es PE ae she tale Rat 2 48 

LSivitish..s dias | ios el: OS 4Bp arial esi 
Satlees 2 +60 5° 42’ 

Datasets ot oe cit Oa Oe ee 

SBT car jet ieHie. 3% 2 30 

et) SEED OR Ae 0° 30’ 

. Ea OH +75 4° C. Rjight. 

13 © _ 








A 








Calc. Bg. 


N1°26’ E 


1-407 66 
432.6 
840.0 





Needle. 


N1°30’E 


N32°10’W | N32°00’W 


The Needle reading, as a rough check, may be recorded 
on the right-hand page if preferred. The position of the 
Vertex, length of Tangent and length of curve may appear 
in any vacant space. Double the last deflection on each 
long chord equals the angle subtended by it, and the sum 
of the angles equals A. The form given is for the second 
method of deflections and also applies to a compound curve. 


A slight change will be neces- 
sary for the first method. 
110. The stations on a curve 
may be located by deflec- 
tions only, without linear 
measurements. For this pur- 
pose two transits are set at 
two transit points on the 
curve, as A and B, Fig. 7, and 
the proper deflections for any 
station are made with both 
instruments, the station being 





Fras 7. 


located by finding the intersection of the two lines of colli- 


mation. " 


62 FIELD ENGINEERING 


This method requires that the two transit points shall 
have been previously established, that their distance from 
each other shall be known, that they shall be visible from 
each other, and that they shall both command a view of the 
stations to be located. It is not therefore generally useful, 
but may be resorted to when certain stations fall where 
chaining cannot be accurately done, as in water or swamps. 
The chord joining the two transit points becomes, in fact, 
a base-line, and the deflections form a series of triangula- 
tions. 

111. Metric Curves. In the Metric System the station 
length is 20 meters, or 65.617 feet. The consecutive station 
stakes are marked in even numbers only, so that ten times 
a station number gives the length of line to that point in 
meters. 

Metric curves, laid out with 20 meter chords, are rated 
and identified, not by the central angle per station, D, but 
by the deflection angle per station, dm, so that what the 
French call a 1° curve has, in fact, 2° of central angle per 
station, If Ry denote the metric radius, then 


10 


sin dm 





(16a) 


from which all other parts of the metric curve may be cal- 
culated by the usual formulas, in metric units. Or we may 
derive the same values from our own tables. Thus, taking 
any value of dm; Rm = 0.2R for D = 2dm, and on the same 
line of the table, tm = 0.2t and mm = 0.2m, the tangent 
offset and middle ordinate of a single chord of 20 meters. 

A close approximation to the value of other parts may 
be had by observing that, for any given value of the central 
angle, A, and D(= 2d), the tables give 7’, or C, or M, which 
multiplied by 0.2 produces 7'm or Cm or Mm. 

To find the equivalent curve; reduce Rm to feet, and from 
Table I take the corresponding value of D, 


SIMPLE CURVES 63 


C. Location of Curves by Offsets 


112. A curve may be located by linear measurement 
only, without angular deflections.. There are four general 
methods, viz.: 

By offsets from the chords produced, 

By middle-ordinates, 

By offsets from the tangents, and 

By ordinates from a long chord. 


To locate a curve by offsets from the chords 
produced. , 


When the curve begins and ends at a station. 
113. Let A, Fig. 8, be the P.C. of a curve taken at a station, 
to locate the other stations, a, b, c, etc. The chords Aa, 





Fia. 8. 


ab, bc, etc., each equal 100 feet, and since the angle AOa = D, 
the angle VAa = 3D. (Table XLI, 20.) Taking an offset 
ax = t, perpendicular to the tangent, we have in the right- 
angled triangle Aza. 
ax = Aa X sin 3D 
or , 
t = 100 sin 4D (34) 


The offset ¢ is called the tangent offset, and its value is 
given for all degrees of curve in Table I, col. 4. 


64 FIELD ENGINEERING 


If the curve were produced backward from A, 100 feet 
to station z, the offset zy would equal t; and if the chord 
zA were produced 100 feet from A to a’, the offset a’x would 
also equal t. Therefore the distance aa’ = 2t, and the angle 
aAa’ = D. So if we produce the chord Aa 100 feet to 0b’, 
the distance bb’ = 2t. 

To lay out the curve, stretch the tape from A, keeping the | 
forward end at a perpendicular distance, t, from the line 
of the tangent to locate station a. Then find the point 
b’ by stretching the tape from a in line with a and A, and 
then stretching the tape again from a, fix its forward end 





& 


rer y Ye te 
= 


at a distance from b’ equal to 2¢. This gives station 6. In 
the same way find other stations. 

When the last station, as d, of the curve is reached, pro- 
duce the curve one station farther to e’’.. Then the tangent 
through d is parallel to the chord ce’’, and laying off ¢ from 
c and e”’ perpendicular to this chord, the tangent c’’e is 
found. If the work has been correctly done the tangent 
ce will coincide with the given tangent VB. 

When the curve begins and ends with a subchord. 

114. Let A, Fig. 9, be the P.C. and Aa the first sub- 
chord = c, and the angle VAa = 3d, and let the offset 
ax = k. Then 


t; = c sin 4d (35) 


Producing the curye backward to the nearest station 2, 


SIMPLE CURVES 65 


we have another subchord Az = (100 — c), and the angle 
yAz = 3(D — d), and putting the offset yz = t 


t, = (100 — c) sin 3(D — d) (36) 


Laying off the two subchords on the ground, and making 
the proper offsets, 4 and tz, at the same time, we fix the 
position of the two stations a and z on the curve; after which 
we may produce the chord za 100 feet to b’, and proceed as 
before until the curve is finished. 

If the curve ends with a subchord, as dB, produce the 
curve to the first station beyond B, as e’’, then calculate 
the two offsets for the two subchords Bd and Be’’, and lay 
them off from d and e” perpendicular to the supposed 
direction of the tangent. If the line d’’e so obtained coin- 
cides with the given tangent, VB, the work is correct. 

115. We may find the values of t; and t. otherwise than 
by the formulas above, for in Fig. 8 we have shown that the 
angle aAa’ = aOA, and since these triangles are isosceles, 
they are similar; therefore 


Fig. 8, OAs Aa: AG: ‘aa? 
. R : 100 :: 100 : 2¢ 
_ (100)? 


ES di EY (37) 
and similarly, Fig. 9, i 
; C , 

ty oR (38) 

Hence 
besiitiec? <(100)* 
c*t 

“5 i = (100)? (39) 


Thus t; may be found by multiplying the square of the 
subchord by the value of ¢ given in Table I, and dividing 
the product by 10,000. As c is always less than 100, so 
is always less than ¢. 

116. In eqs. (35), (38), and (89) it is customary to use the 
nominal values of c, and this can produce no error in ti 
exceeding 0.005, when the degree of curve does not exceed 
10°. In the case of a very sharp curve, formula (41) is 
preferable. 


54 


66 FIELD ENGINEERING 


To locate a curve by middle-ordinates. 


When the curve begins and ends at a station. 

117. In Fig. 10, let A be the P.C. at a station, and let 
a and z be the next stations on the curve either way from A. 
Then, since zy = ax =t, the chord za is parallel to the 
tangent AV, and Ag =t. Hence, having any two con- 
secutive stations on the curve, as z and A, we may lay off 
the tangent offset ¢ from A to g on the radius, and find the 
next station, a, 100 feet from A on the line zg produced. 
Then laying off ah = t on the radius aO, a-point on the line 
Ah produced and 100 feet from a will be the next station b. 





Fie. 10. Hier 11. 


On reaching the end of the curve, the tangent is found 
precisely as described in the method by chords produced, 
S113. 


: ; OE OL 
In Fig. 10, since vers AOa = Oa o we have 
t = & vers D (40) 


When the curve begins or ends with a subchord. 
118. Let A, Fig. 11, be the P.C., and a and z the nearest 
stations. Then Aa =c, the first Babehond! and aQA = d, 
and by analogy, we have from the last equation, if ax = 
and zy = te 
t; = R vers d 
to = FR vers (D — d) } 


or eq. (39) may be used if preferred. 


(41) 


* 


SIMPLE CURVES 67 


Having found the two stations, a and z, on the curve, 
lay off from the forward station a, ah =t on the radius, 
and so continue the curve as described above. 

When the end of the curve is reached, produce the curve 
to the next station beyond, and find the tangent by offsets 
as described in the previous method, § 114. 


To locate a curve by offsets from the tangents. 


When the'curve begins at a station. 

119. Let A, Fig. 12, be the P.C. at a station. Then the 
next station a is located by the tangent offset ¢, taken from 
Table I, or calculated by eq. (40). To calculate the dis- 





tances and offsets for the following stations, 6, c, etc., in the 
diagram draw lines through the points b, c, etc., parallel to 
the tangent AV, intersecting the radius AO in g’, g’’, etc., 
and draw the lines bz’, cx’’, etc., perpendicular to the tangent. 
Then 

Az’ = g'b = Ob sin bOA 


or 
Az’ =Rsin 2D . 
Az” =R sin 3D (42) 
and ete. ete. 
Also, 


,. ba’ = g’A = Ob vers bOA 


68 FIELD ENGINEERING 


or t’ = R vers 2D 
t= Rivers 3D (43) 
and ete. ete. 


But these calculations may be avoided, for as twice ag 
equals the chord of two stations, so twice bg’ equals the chord 
of four stations, and twice cg’’ the chord of six stations, 
etc. So‘also as Ag is the middle-ordinate of two stations, 
Ag’ is the middle-ordinate of four, and Ag’ the .middle- 
ordinate of six stations, ete. Hence the rule: ; 

The distance on. the tangent from the tangent point to the 
perpendicular offset for the extremity of any arc is equal to 





Bic. 13, 


one half the long chord for twice that arc; and the offset from the 
tangent to the extremity of piel are is equal to the middle-ordinate 
of twice that arc. 

The long chords and middle-ordinates may be taken from 
Tables IV and VI for 2, 4, 6, 8, etc., stations, when the 
P.C. is at a station, or for 1, 3, 5, 7, etc., stations, when the 
P.C. is at + 50, or half a station. 

If the offsets from the first tangent AV prove inconven- 
iently long, the second half of the curve may be located from 
the other tangent BV, beginning at the point of tangent B, 
and closing on a station located from the first tangent. 

When the curve begins with a subchord. 

120. If d = the angle at center, subtended by the first 
subchord, we have for the distances on the tangent (Fig. 13) 


SIMPLE CURVES | 69 


Ax =Rsind 
Az’ =Rsin (d+D) 


Ax =. R&in (POD) aa 
etc. etc. 
and for the offsets (Fig. 11) 
4 = Rversd 
t' = R vers (d+ D) (45) 


t’’ = R vers (d + 2D) 
etc. etc. 


/ if the first subchord equals 50 feet (nominal), then d = 4D, 
and the Tables IV and VI may be used as explained above. 





Fig. 14. 


These tables may be used in any case, by adopting a tem- 
porary tangent through any station, and laying off the 
distances on this, and making the offsets from it. 

When a curve is located by offsets the chain should be car- 
ried around the curve, if possible, to prove that the stations 
are 100 feet apart, — 


To locate a curve by ordinates from a long 
chord. 


When the curve begins and ends at a station. 

121. In Fig. 14 draw the long chord AB, joining the 
tangent points, and from this draw ordinates to all the sta-: 
tions on the curve. We then require to know the several 
distances on the long chord Aa’, a’b’, b’c’, etc., and the 
length of ordinate at each point. 


70 FIELD ENGINEERING 


Let C = the long chord AB, then eq. (22) 
| C =2R sin 34 


If ais the second station and 7 next to the last on the curve, 
join ai, and let the chord az = C’. Then since the are 
Aa = ik = D, the angle at the center subtended by C’ 
is (A — 2D). 
; C’ = 2R sin 4(A — 2D) 


Again, if we join 6 and h the next stations and let bh. = C” 
C”’ = 2R sin 3(A — 4D) 
and so on for other chords. 
Since Aa’ = ki, C = C’ + 2Aa’ 
i al Se 
WE 2 





: Aa’ 
Similarly, “ 
(Ol 2 sabe dtd 

») a 


—_ 


OD = 





Thus we continue to find the distances up to the middle 
of the curve, after which they repeat themselves in inverse 
order. 

122. When the long chord C, subtends an even number of 
stations (as 10 in Fig. 14), the middle-ordinate of the chord 
is the ordinate of the middle station, as e. Since the chords 
AB and ai are parallel, the ordinate a’a or 7’i is evidently 
equal to the difference of the middle-ordinates of these 
chords. 

Let M, M’, M”, ete., be the middle-ordinates of the 
chords C, C’, C’’, ete. Then eq. (23) 


M =RversiA 
M’' =R vers 3(A — 2D) 
M" = R vers 3(A — 4D) 


etc. ete. 
And aa =ivi = M — M’ 
b’b = h’h = M — M"” 
etc. etc, etc, 


The values of the chords and middle-ordinates may be 
taken at once from Tables IV and VI. 


SIMPLE CURVES 71 


Example.— It is required to locate a 4° curve of ten stations 
by offsets from the long chord. 
By Table IV: 


Diff. 3Diff. 








gee nies e800. OSs nm oe Sine dE s10n = da’ Ske 
8 Cc! = 789.803 nee le 
i: tt 194.059 | 97.030 =a’b' =Wh 
6 co" — 595.744 7 MO 
eG: C _ 398 789 196.962 | 98.481 = b’c’ =h’g 
se ‘iv 198.904 | 99.452 = c’d’ = g’f' 
“ OE ON tog aa! | GN O80 deh oe ek 
0“ | CY = 000.000 Pe Mat Tite walisiod ain 





From Table Wii 


Diff. : 
Bees UNECE pte 80. 002 =a'a = 
8 M' = 55.500 ; 
be oil : 55.094 =b'b = f’h 
6 M" = 31.308 
‘4 pil 72.459 =.clce =9'g 
4 M" = 13.943 
‘@ copie @ 82.912 =d'd=f'f 
2 MY = 3.490 sae ee 
Camere eM wit: 000 


123. When the long chord C subtends an odd number of 
stations, the middle-ordinate will fall half-way between two 
_ stations, and need not be laid off. 

If the ordinates near the middle of the curve prove incon- 
veniently long, we may subtract M — M’, M’ — M”, etc., 
and so obtain in Fig. 14 a’a, b’’b, c’’c, etc. We then lay 
off Aa’, a’a, ab’, b’’b, be’, ete., turning a right ange at 
every point. The chain should be carried along the curve 
at the same time to make the stations 100 feet apart. 

Example.—It is required to locate a 10° curve of nine 
stations by offsets from the long chord. 


By Table IV: 
Diff. $Diff. 

eee 153.209 76.604 = Aa’ 
7“ 658.105 Y 

i | 173.205 86.603 = a’b 
5 “© 484.900 
he SA 196.962 98.481 
Sera ngs 100.000 50.000 





Te FIELD ENGINEERING 


By Table VI: 
Diff. 
I 

9 sta. 168.029 64.279 idee 

7 ‘© 103.750 . 
50. 000 = tpt 

5 * 53.750 - 
34,202 =c¢''C 

BAS SL OLDS 

of 17.365 ete. 
1 2.183 2 183 
0“ 0.000 








124. The tables can be used equally well when the curve 
both begins and ends, with a half station; also to locate 
half-station points throughout the curve, but in the latter 





Fiaueto: 


case the numbers are taken from consecutive columns of the 
tables instead of from alternate columns, as in the above 
examples, 

When the curve begins or ends with any subchord. 

125. Let A, Fig. 15, be the P.C. and Aa = c« the first 
subchord, and d the angle it subtends at the center. In 
the diagram draw the long chord AB, and the ordinates 
to each station, and through each station draw a line parallel 
to AB, and let AOB = A. 

Since the angle VAB = 3A and VAa = 3d, the angle 
aAB = 3(A —d). The deflection angle from the sub- 
chord Aa produced to the chord ab is 3(d + D), the de‘lec- 
tion angle between any two consecutive chords of 100 feet 
is 3}(D+ D) =D. Therefore the angle 


SIMPLE CURVES 13 


bab” = 3(A — d) — 3(d + D) = 3(A — 2d — D) 

che’ = 3(A — 2d — D) — 3(2D) = 7(A — 2d — 3D) 

edd!’ = 3(A — 2d — 3D) — 3(2D) = 3(A — 2d — 5D) 
etc. etc. etc. 


Solving the several right-angled triangles we have, Fig. 15. 


Aa’ =c. cos 3(A —d) 
ab’’’= 100 cos 3(A — 2d — DPD) 

- be’’ = 100 cos (A — 2d — 3D) (46) 
dd’’ = 100 cos 3(A — 2d — 5D) 
etc. etc. 

And also 

a’a=c. sin $(A — d) 
b’’b = 100 sin 3(A — 2d — D) 
c’’c = 100 sin 3(A — 2d — 3D) (47) 


d’’c = 100 sin $(A — 2d — 5D) 
etc. etc. 


When the middle point of the curve is passed the minus 
quantities in the parentheses become greater than A, making 
the parentheses negative, and, therefore, the sines negative, 
and indicating that such values as are determined by them 
must be laid off toward the long chord AB. 

By a proper summation of the quantities determined by 
eqs. (46) and (47) we obtain the distances Aa’, Ab’, Ac’, 
-etc., and the ordinates a’a, b’b, c’c, etc., and the curve may 
be located accordingly. It is well to make all the necessary 
calculations before beginning to lay down the lines on the 
ground, thus avoiding confusion and mistakes. 

Example.—The P.C. of a 3° 20’ curve is fixed at + 25 feet 
beyond a station, and the central angle is 16° 24’ = A. 
It is required to locate the curve by ordinates from the long 
chord. 

We havec = 100 — 25 = 75andd = 2°30’ and D = 3° 20’. 
Hence, eqs. (46) 


Aa’ = 75 cos B57 = 74.449 74.449 = Aa’ 
ab’’ = 100 cos 4° 02’ = 99.752 174.201 = Ad’ 
be’’ = 100 cos 0° 42’-= 99.993 274.194 = Ac’ 
d’’d = 100 cos (— 2° 38’) = 99.894 374.088 = Ad’ 
e’’e = 100 cos (— 5° 58’) = 99.458 473 *546"= Ae 
e’'B = 17 cos (— 7° 55’) = 16.838 490.384 = AB 


74 FIELD ENGINEERING 


_ By eqs. (47) 
a‘gi== 75,8 86° 571. 9-075 1> 9.075 =a'a 
b’’b = 100 sin 4°-O2 (i 7.034 | 16.109 = b’b 
c’c = 100 sin Q*. 4240 = By2228) li wel =e 
ed’’ = 100 sin (—2° 38’) = — 4.594 | 12.737 =d’d 
de’’ = 100 sin (— 5° 58’) = — 10.895 | 2.342 = e’e 
ee’ = 17 sin (—7° 55’) = — 2.341] 0.000 


The same formulas can be used when the curve begins at 
a station by makingc = 100 and d = D. 

126. The methods of locating curves by linear measure- 
ments do not require the use of a transit, although one may 
be used to advantage for giving true lines, turning right 
angles, etc. When a transit is not used the alinements 
should be made across plumb-lines suspended over the exact 
points previously marked on top of the stakes. A right 
angle may easily be obtained, without an instrument, 
by laying off on the ground the three sides of any of the 
right-angled triangles represented in the following table 
(or any multiples of them), always making the base coin- 
cide with the given line. 


TABLE OF RIGHT-ANGLED TRIANGLES 


Base. Hypothenuse. Perpendicular. 

4 5) 3 
12 13 5) 
20 29 21 
24 25 7 
40 41 9 
60 61 11 
84 895 13 


D. Obstacles to the Location of Curves 


127. To locate a curve joining two tangents when the in= 
tersection V is inaccessible. Fig. 16. 

From any transit point p on one tangent run a line pg to © 
intersect the other tangent; measure pg and the angles 
it makes with the tangents. Then the sum of the deflec- 
tions at p and q equals the central angle A. Solve the triangle 
pqV and find Vp. Having decided on the radius R of the 
curve, calculate the tangent distance VA by eq. (21), and 


4 


SIMPLE CURVES 15 


lay off from p the distance pA = VA — Vp to locate the 
point of curve. The point p being assumed at random, 
Vp may exceed VA, in which case the difference pA is to be 
laid off teward V. 

In case obstacles prevent the direct alinement of any line 
pq, it may be replaced by two or more lines, and the distances 
Vpand Vqfound. There will be as many triangles to solve 
as there are courses. The algebraic sum of the several 
deflections will equal A. 

128. To locate a curve when the point of curve is in- 
accessible. Fig. 17. 





SK] 










mS 
SS 
~TAASSS 









WN 


S 





& 


igieel, 17/, 


Assuine any distance Ap on the curve which will reach to 


an accessible point p. Then by eq. (19) the angle 


From the figure, 
Ap’ =Rsin pOA 


p'p = R vers pOA 
Vp' =VA — Ap’ 


Measure Vp’ and p’p to locate a transit point at p; and meas- 
ure an equal offset from some transit point on the tangent, 
as qq’. This gives a line pq’, parallel to the tangent, from 
which deflect at p an angle equal to pOA for the direction 
of a tangent through the point p. 


Instead of measuring the second offset gq’ we may deflect 


/ 


from pq an angle found by tan qpq’ = a and so obtain 


76 FIELD ENGINEERING 


the line pq’ parallel to the tangent. Or we may deflect 
from pV the angle found by tan pVp’ = aa to obtain 
the line q’p produced, from which the tangent to the curve 
at p is found as above. 


Again, we may lay off from V, the external distance Vh 
found by eq. (24) or Table III on a line bisecting the angle 
AVB. This gives us h, the middle point of the curve, and a 
line at right angles to AV is tangent to the curve at h, from 
which the curve may be located in either direction. 

129. To locate a curve when both the Vertex and Point 
of curve are inaccessible. Fig. 18. 





Fig. 18. Fia, 19, 


From any point p on the tangent run a line pq’ to the other 
tangent, and so determine pA as in § 127. Suppose the 
curve produced backward to p’ on the perpendicular offset 


/ 


pp’. 

Then 

sin p‘OA = Be and pp’ = R vers p’/OA 

Having located the point p’, a parallel chord p’qg may be 
laid off, giving a point g on the curve, since p’g = 2 X pA. 
At g deflect from gp’ an angle equal to p’OA for a tangent to 
the curve at q. 

If any obstacle prevents using the chord p’q, any other 
chord as p’s may be used, by deflecting from p’q the angle 
gp’s = 4(qOs) and laying off its length, 


p's = 2R sin (p'OA + qp’s) 


SIMPLE CURVES ries 


At s a deflection from the chord sp’ of (p’OA + qp’s) will 
give the tangent at s. 

If obstacles prevent the use of any chord, the methods 
described in § 131 may be resorted to. 

130. To pass from a curve to the forward tangent when the 
Point of Tangent is inaccessible. Fig. 19. 

From any transit point p on the curve, near the end of 
the curve, run a chord parallel to the tangent. The middle 
point g of the chord will be on the radius through the point 
of tangent 8B. At any convenient point beyond this an 
offset equal to pp’ = R vers pOB may be made to the 
tangent, and at some other point an equal offset will fix 
the direction of the tangent. 

Otherwise, if an unobstructed line pq can be found inter- 
secting the tangent at a reasonable distance from B, measure 
the angle q'pq = pgp’, and lay off the distance 


; pp’ 


~ sin g’pg 





Pq 
to fix the point g. Then . 
Bg = p’g — p'B = pp’ cot q'pq — F sin pOB. 


Otherwise, assume an arc of any number of stations from 
p to q’” on the curve produced, and take the length of chord 
from Table IV. Lay off pq’’, and from q’” lay off q’q = R 
vers g’’OB, perpendicular to the tangent, to locate g. The 
angle pq’’¢ = 90° — q’pq’’, and 
the distance gB = R sin q’OB. 

131. To pass an obstacle on 
a curve. Fig. 20. 

From any transit point A’ on 
the curve take the direction of 
along chord which will miss the » 
obstacle, as A’B’. The length 
of this chord is 2R sin V’A'B’, 
V’'A’ being tangent to the curve 
at A’ (see eq. (22)), and by Fig. 20. 
measuring this distance, the 
point B’ on the curve is obtained. If the angle V’A’B’ is 
made equal to the deflection for an exact number of sta- 
tions, the chord may be taken from Table IV, 





78 FIELD ENGINEERING 


If the chord which will clear the obstacles would be too 
long for convenience, as A’g’, we may measure a part of 
it as A’p’, and then, by an ordinate to some station, regain 
the curve at p. The distance on the curve from A’ to p 
being assumed, the distances A’p’ and p’p are calculated 
by the methods given in §121 to §125. If p’p can be 
made a middle-ordinate the work will be much simplified. 
If more convenient the middle-ordinate may first be laid 
off from A’ to p’’, and the half chord afterwards measured 
from p’’ to locate p. 

Again, we may calculate the auxiliary tangent A’V’ for 
any assumed length of curve A’B’, and lay off the distance 
A'V’ and V’B’, deflecting at V’ an angle equal to twice 
V’A’B. But if the point V’ should prove inaccessible, we 
may conceive the auxiliary tangents to be revolved about 
the chord A’B’ as an axis, so that V’ will fall at V’’, and 
the lines A’V” and V’’B’ may be laid out accordingly. If 
these in turn meet obstructions, we may run a curve from 
A’ to B’ of same radius as the given curve, but tangent 
to A’V". and VB". 

Again, the entire curve or any portion of it may be laid 
out by offsets from the tangents, or by ordinates from a long 
chord, as already explained, § 119 to § 126. 

In case any distance on a curve must be measured. by 
triangulation, as in crossing a stream, a long chord may be 
chosen, either end of which is accessible, and the triangula- 
tion is then performed with respect to this chord or a part 
of it, as upon any other straight line. 


SPECIAL PROBLEMS IN SIMPLE CURVES 


132. Given: a curve joining two tangents, to find the change 
required in the radius R, and external distance E, for an 
assumed change in the value of the tangent distance T. Fig. 21. 


Let T=AV=VB and T’=A'V = VB’ 
R = AO R' =A’0’, 
E=VH E' = VH’ 


Then 7 — T’ = AA’ = the given change. 


SIMPLE CURVES 79 


By eq. (25) R=T cot 4A 
R’ = T’ cot 4A 
OG = R — R’ = (T — T") cot 1A (48) 


By eq. (26), similarly, 
HH’ = E — E’ = (T — T’) tan 4A (49) 


Kas. (48) (49) give the changes 
in R and E-for any change in T. 
When T is increased R and FE 
will be increased also, and vice 
versa. 

Example.—A 4° curve joins two 
tangents, making an angle of 
38° = A, and it is necessary to 
shorten the last tangent distance 
80 feet. What will be the 
change in the radius and in the 








external distance? Oh 
Eq. (48). | T — T’ = 80 log 1.903090 
1A 19° log cot 0.463028 
Ans. R — k’ 232.34 log 2.866118 
R 1432.69 
Rs 1200.35 or about 4° 46’ = D’, 


If the tangent distance had been increased 80 feet we should 
add the above to R. 


Rk’ = 1665.03 or about 3° 26’ = D’ 


Eq. (49) T — T’ =80 log 1.903090 
ah 9° 30’ — log tan 9.223607 
Ans. E — E’ ~~ 13.387 log 1.126697 


133. Given: a curve joining two tangents, to find the change 
required in the radius R, and tangent distance T, for any 
assumed change in the value of the external distance KE. Fig. 21. 

We suppose HH’ given, to find OG and AA’, 


By eq. (24) E = R ex sec 3A 
», FE’ =R’' exsec $A 


80 FIELD ENGINEERING 


a ean 
heh ear pt Mreeaas (50) 
By eq. (49) 


AA’ =T —T’ = (& — E’) cot $A (51) 


Example.—A 4° curve joins two tangents, making an 
angle of 38° = A, and it is necessary to bring the middle 
point of the curve 25 feet nearer the vertex V. . What changes 
are required in the radius and point of curve? 

Ans. From eq. (50), R — R’ = 483.87; then R’ = 998.82 
and D’ = 5° 44’ about. 

From eq. (51), T — T’ = 149.39. 

But if the point H, Fig. 21, were to be moved 25 feet 
further from the vertex V, then 


Rk’ = 1866.56 or about 3° 04’ = D’ 


and the P.C. will be moved 149.39 feet further from the 
vertex. 

It is preferable to assume some radius from Table I near 
the value of R’ found as above, and from this calculate the 
value of T’ by eq. (21). 

134. Given: a curve joining two tangents, to find the change 
made in the tangent distance T, and external distance E, by 
any assumed change in the value of the radius R. Fig. 21. — 

By eq. (48) } 

AA’ =T — T’ = (R — R’) tan 3A (52) 

By eq. (50) 

HH' = E — EK’ = (R — R’) ex sec 3A (53) 


The changes calculated by eqs. (52) (53) will be added to 
or subtracted from 7 and E respectively, according as the 
radius is increased or diminished. 

135. Since for a constant value of the central angle A, 
the homologous parts of any two curves are proportional to 
each other, we may write at once 


(54) 


SIMPLE CURVES 81 


136. Given: a curve joining two tangents, to change the 
position of the Point of curve so that the curve may end 
ina parallel tangent. Fig. 22. 

Let AB be the given curve, AV, VB the tangents, and 
V’'B’ the parallel tangent. Then VV’ is the distance from 
one vertex to the other; and since there is no change in the° 
form or dimensions of the curve, we may conceive it to be 
moved bodily, parallel to the line AV, until it touches the 
line V’B’, when every point of thé curve will have moved 
a distance'equal to VV’. Hence AA’ = OO’ = BB’ = VV’. 
Therefore, run a line from B parallel to AV, intersecting the 
new tangent in B’, measure BB’, and lay off the distance 





Fig, 22. Fre, 23, 


from A to find A’. In the figure the new tangent is taken 
outside the curve, and so A’ falls beyond A, but if the new 
tangent were taken inside the curve at VB’, the new . 
P.C. would fall back of A at some point A”. 

If the parallel tangent is defined by a saben dicts offset, 
from B, as Bp, since the angle BB’p = A 


Bp 


Euratom A 





(55) 


137. Given: a curve joining two tangents, to find the 
radius of a curve that, from the same Point of curve, will 
end in a parallel tangent. Fig. 23. 

Let AB be the given curve, AV, VB the tangents, anal 
V’B’ the parallel tangent; and let AO = R and AO’ = R’. 

Since the central angle A remains unchanged, the angle 
1A between the tangent and long chord remains unchanged; 


&2 FIELD ENGINEERING 


therefore V’AB’ = VAB, and the new point of tangent 
is on the long chord AB produced. Find on the ground 
the intersection of V’B’ with AB produced and measure 
BB’. Inthe diagram draw Be parallel to AO, then BeB’ =A, 
and by eq. (22) 

BB’ = 2Be sin 3A 


but 
Be = OO’ = Rk’ —R 
na tp ee? ay 
4 aaa eT IN , (56) 


Use the lower sign when the parallel tangent cuts the 
chord AB at some point B’’, since R’ will evidently be less 
than R. 

If the parallel tangent is defined by a perpendicular offset, 
as Bp = B’f; since BeB’ = A 


Bp = Be vers A = (R’ — R) vers A 


Bp 
vers A 





R’=R+ (57) 
Add or subtract as explained above. 

If the long chord C = AB is known, then the new long 
chord C’ = AB’ or AB” =C + BB’, and by eq. (54) 

[omg ee (58) 
C 

138. Given: a curve joining two tangents, to change the 
radius, and also the Point of curve, so that the new curve 
may end in a parallel tangent directly opposite the 
given Point of tangent. Fig. 24. 

Let AB be the given curve, AV, VB the tangents, V’B’ 
the parallel tangent, and B’ the given tangent point on the 
radius OB produced. 

In the diagram, produce the tangent AV and the radius 
OB to intersect at K. Then 


BK = RexsecA 


B’'K = R’ exsecA 
Subtracting we have 
B’ = (R — R’) exsecA 


SIMPLE CURVES 83 


BB’ 


R— Fk =——— 
ex sec A 


(59) 
from which #’ is easily determined, as in §§ 132 and 133. 

To find the change AA’ of the P.C., in the diagram draw 
O’G parallel to A’A; then 


O’G = OG tan A 
or 
AA’ = (R —R’) tan A (60) 





Hr@. 25. 


By substituting the value of (R — R’) from eq. (59) and 
observing Table XLII, 42, we have 


PAPAd ea 1513 x COU aN (61) 


Observe that eqs. (59), (60), and (61) may be derived directly 
from eqs. (50), (52), and (51) respectively by writing A for 3A. 

139. Given: a curve joining two tangents; to find the new 
tangent points after each tangent has been moved 
parallel to itself any distance in either direction. Fig. 25. 

Let A and B be the given tangent points, and A’ and B’ 
the new tangent points required. Let the known perpen- 
dicular distances Ag = a, and Bp = b. We then require 
the unknown parallel distances gA’ = x and pB’ = y. 

Since the form and dimensions of the curve remain un- 
changed we may conceive the curve to be moved bodily 
into its new position on lines parallel and equal to the line 
VV’ joining the vertices. Then AA’ = OO’ = BB’ = VV’. 
In the diagram draw VK parallel and equal to Bp = 6 and 


R4 FIELD ENGINEERING 


V’'H parallel and equal to Aq =a. Then VH = qA’ =2, 
and V’K = B’p = y. Since VGV’ = A, we have 


a 





G = in Db and GH = 
sin A 





tan A 
and since 
VH = VG—-GH =x 
g thant Oe 
sin A. stars 
Similarly (62) 
b a 








: ¥tanA sind 

When the new tangents are outside of the given curve, 
the offsets a and b are considered positive; if either new 
tangent were inside of the given curve its offset would be 
considered negative. In solving eqs. (62) if « and y are 
found to be positive they are to be laid off forwards from 
q and p, as in Fig, 25; if either is found to be negative it 
is to be laid off in the opposite direction. 

Example.—A certain curve has a central angle of 50° = A, 
and it is proposed to move the first tangent in 20 feet and 
the second tangent out 12 feet. Required, the distances 
on the tangents from the old tangent points to the new. 








Fig. 26. 
Herea = — 20 andb = + 12 

+b 12 15070181 logs OD pec 1.301030 

A 50° log sin 9.884254 | A 50° log tan 0.076186 
15.665 1.194927] — 16.782 1.224844 

2 = 15.665 — ( — 16.782) = + 32.450 

abe 12st 1.079181 | —a 20 1.301030 

A 650° log tan 0.076186 | A 50° ~=log sin 9.884254 
10.069 1.002995/ — 26.108 1.416776 


y = 10.069 — (— 26.108) = + 36.177 


; {a = — 32.450 
For + a and — b ly = — 36.177 
{x= —- 1.120 


For +a and +b ly = — 15,989 


SIMPLE CURVES 85 


fx=+ 1.120 


For —aand —} Ly = 415.939 


If we have a and «x givin to find b and y: Solving eqs. (62) 
for 6 and y we obtain 


b=xsn4+acosA \ (63) 


y=xcosA—asinA 


In which the algebraic signs of the quantities must be 
observed as above. 

140. Given: a curve joining two tangents, to find a new 
Radius and new position of the Point of curve, such that 





ENT Gea 2iake 


the curve may end at the same point as before, but with a given 
change in the direction of the forward tangent. Fig. 27. 

Let AB be the given curve, AV, VB the given tangents, 
V’B the new tangent, and VBV’ the given change in direc- 
tion. Let A’ =A+BVV’". 

In the diagram draw BG perpendicular to AV produced; 
then . 





BG = R vers A 
=R’vers A’ 
Hence 
jeg pversA 
soni vers A’ | seg! 
and 
AA’ = AG — A’G = R sin A — R’ sin A’ (65) 


In the figure the change in direction of tangent makes A’ 
greater than A} therefore V’ falls beyond V, and A’ beyond 


86 FIELD ENGINEERING 


A; but if the change made A’ less than A, then V’ and A’ 
would fall behind V and A respectively, and R’ would be 
greater than R. 

The same formulas apply to the converse problem in 
which B is taken as the point of curve, and A and A’ as 
points of tangent. 

141. Given: a curve joining two tungents, to find the change 
in the Point of curve when the forward tangent takes a new 
direction from the vertex, V. Fig. 28. 

By eq. (21) 


VA.=R.tan 4A, VA’ = R tan 3A’ 
AA’ = R (tan $A — tan 3A’) (66) 





Oo’ 
Fic. 28. | Fig, 29, 


142. Given: a curve joining two tangents, to find the new 
radius, R’, when the forward tangent takes a new direc- — 
tion from the vertex, V. Fig. 29. 

By eqs. (21) (25) 


VA = Rtan 3A, R’ = VA cot $A’ 
R’ = R tan 4A cot 3A’ (67) 


143. Given: a curve joining ‘two tangents, and a given 
change in the direction of the forward tangent from the 
vertex, to find the radius and point of curve of a curve 
that shall pass at the Same distance, VH, from the 
vertex. Fig. 30. 

Let AB be the given curve, BVB’ the given change in 
direction of tangent, and VH’ = VH. Let A’ = A+ BVB’, 
then eq. (24) 


SIMPLE CURVES 87 


VH = Rex sec 3A = VH' = R’ ex sec 4A’ 


pO secnsA 


Ir v= Rh eae is (68) 
By eq. (28) 
VA = VEH cot {A, VA! = VH’ cot 3A’ 
AA’ = VH (cot {A — cot 1A’) (69) 


But in case A’ = A — BVB’, AA’ becomes negative and 
must be laid off backward from A. 

Example.—Given a 2° curve, A = 80° and BVB’ = — 10° 
* A’ = 70°, 





Fia, 30, Fie, 31. 


Ans. From eq. (68), D’ = 1°27’ nearly, and from eq. 
(69), AA’ = — 371.08, which means that A’ must be laid off 
backward from A. 

144. Given: two indefinite tangents, a point. situated be- 
tween them, and the angle A, to find the radius R, and tan- 
gent distance 'T of a curve joining the tangents which shall pass 
through the given point. Fig. 31. 

If the given point is on the bisecting line VO, as H, meas- 
ure VH = E, and find R and T as in §§ 97, 98. 

When the given point, as P is not on the bisecting line VO; 
if a line GK is passed through P perpendicular to VO, it will 
be parallel to any long chord, as AB, and the angle VGK = 3A. 
The curve passing through P will intersect GK in some other 
point P’; the line GK is bisected by the line VO at J, and 


PI = P'l, 


88 FIELD ENGINEERING 


If the given point P is located by a perpendicular offset 
from the tangent, as PL; in the triangle PLG, LG = PL 
cot 3A. Lay off LG, and at G deflect VGK = 3A, and 
measure GP and PK. Since by Geom. (Table XLI, 24) 
GA? = GP’ X GP, and GP’ = PK; 


GA = V GP X PK (70) 


Lay off GA; and A is the Point of curve, AV = T, and 
R = AV cot 3A. . 

If the given point were located by an offset from BV, 
find B first, and make VA = BY. 

If the given point P is located by a perpendicular offset 





IP from the bisecting line VO; produce JP to intersect the 
tangent at G and measure PG. Since P’G = GP + 2PI 


GA =. V GP(GP + 2PT) (71) 


whence we have the point of curve A, as before. 

145. Given: a curve, AP, and the radial offset PP’ 
to find a curve which shall pass through the point P’, start- 
ing from the same point of curve A. Fig. 32. 

Let 6 = PP’, and in the diagram draw P’@’ parallel to 
the common tangent AX, and join AP’. Then 


P'G'’ =(R+b5) sina 
GA. = BR — (FR +5) cos A 
tan 3A’ = tan G’P’A’ 
G’ A 


wighoas Wesligeicl oR 
"P@” @adandy 29 a2 


SIMPLE CURVES 89 


P’'G _ (R +b) sind (73). 


'= O'P' = - : 
k 0 sin A’ sin A’ 





When the offset is outward use R + b, when it is inward 
use R — b. 

Example.—Given: a 3° curye of 16 stations and a radial 
offset of 205 feet inward from the P.T. to find the radius of 
the curve passing through the extremity of the offset. 


Ags. 2°= 48” and b = 206/'> Front eg) (72) A" ="G2° 31’ 
and from eq. (73) and Table I., D’ = about 4° OV’. 

If the same offset were made outside of the curve we should 
find R’ log 3.438350, or about a 2° 05’ curve. 

This solution is inconveniently long for ordinary field 
practice. When the offset is small compared with the length 
~ of curve, we may use the following 

Approximate Rule: Divide twice the offset b by the 
length of curve, look for the quotient in the table of nat. 
sines, and take out the corresponding angle, which multiply 
by 100, and divide by the length of curve. The quotient is 
the correction for the given degree of curve; to be subtracted 
when the offset is made outward, and added when the offset 
is made inward. 

This rule is expressed by the formula 

100. , 2b 


D = D+F =~ sin- is (74) 


- 


For the necessary change in A is equal to twice the angle 
PAP’ which is expressed by sin-! * and to reduce this 
. : 100 : 
to Degree of curve we multiply by 7, 2s In eq. (20). 
Whence, eq. (74) results. 
Taking the same example, we have 


DALE ay ian 
7, = sin 14 51 


tat eat 100 _ RESIS 
and correction = 14° 51’ x i600 > +- 0° 56 


Hence D’ =*3° 56’ or D’ = 2° 04’, 


90 FIELD ENGINEERING 


THE VALVOID 


146. Given: any number of circular curves of equal length 
L, all starting from a common point of curve A, in a common 
tangent AX, to find the equation of the curve joining 
their extremities. Fig. 33. 

Let AP be any one of the given curves; 

R = its radius AO; 

D = its degree of curve; 

A = its central angle AOP; 
C = its long chord AP. 





Fia. 34. 


By substituting the value of R from eq. (16) in eq. (22) 
we have 
sin 3A 
sin 4D 
Substituting in this the value of D from eq. (20) and letting 


C = 100 





(75) 


(theta) @ = 3A, (rho) p = — and N = , we have for 


Lb 
100 
the polar equation of the required curve 
sin 6 
p = 





a (76) 
sin 5 
in which p is the radius vector AP, 6 the variable angle 
X AP, the unit of measure is one side of the inscribed polygon 
by which the circular curve AP is measured, and N the num- 
ber of these sides in the length of the curve AP. By the 
conditions of the problem WN is constant, but @ may have 


SIMPLE CURVES 91 


any value whatever. If we let 6 vary from 0° to + 180° 
and from 0° to —180° the point X will describe the curve 
XP’PA shown in the figure, which is called the Valvoid 
from its resemblance to the shell of a bivalve. All circular 
curves tangent to AX at A and having a length L = AX 
will terminate in the valvoid, and the line PP’ joining the 
extremities of any two of them is a chord of the valvoid. 

147. To find a tangent to the valvoid at any point 
P. Fig. 34. See Appendix. 

Differentiating eq. (76) 


l 1 
ee =p (cot-6 tn cot x ' (77) 


which is essentially negative, since p is a decreasing function 
OL 0. 
Let {phi) ¢ = APG, the angle between the radius vector 
and the normal PG. fs 
1 6 
tan ¢@ = N cot Nai cot @ (78) 
The line PK perpendicular to PG is tangent to the valvoid 
at P, and PV perpendicular to PO is tangent to the curve 
Ad? 
Then APV = 6 and VPG =06-— 94, and letting i = 
OPK = VPG, 
A 7=0—g¢=}3A-—6 (79) 


Therefore, to obtain the direction of a tangent to the val- 
void at any point P, deflect from the radius PO an angle 
equal to i = (4A — ¢), on the side of PO farthest from the 
point of curve A. 

The value of 7 may be found by eqs. (78) (79), but we are 
saved this somewhat tedious calculation by the use of Table 


VU, 1, which contains values of the ratio 2 = u for various ~ 


values of A, and length of curve L. Multiplying A by the 
proper tabulated number gives the value of 7 = OPK at 
once; or 

i= (44 — ¢) = ud (80) 


148. To find the radius of curvature of the valvoid 
at any point P. See Appendix. 


92 FIELD ENGINEERING 


Differentiating eq. A 7) we have 
ee & —l—- are 6 cot = + =; : 2 cot? RE 1 
dgz N ¥ w2\" °° 


The general formula for the radius of curvature of polar 


curves 1s 
(0% ta toh 


2 d2p 
pi +250 — Pde? 





T= 


d?p 


do and putting 


Substituting in this the values of p, and 


(F nee — cot 0) = a, we have after reduction, 


WN. ebb 
La): 
a (81) 
® 1 — sya @ cot 9 


This formula being too complicated for convenient use in 
the field, its use is avoided by referring to Table VII, 2, which 
contains values of the ratio < = v for various values of A and 
L. Multiplying the given value of L by the proper tabular 
ratio, gives the value of the radius of curvature of the val- 
void for a short distance either way from the given point P; 
or, 

r=oL (82) 

149. To find the length of are of the valvoid corre- 
sponding to a change of one degree in the value of the angle A. 
Fig. 35. 

From any chord AP suppose a deflection of 1° to be 
made each way to Ap’ and Ap”; then the angle p’Ap” = 
3° = the change in @, and since A = 286, this makes a change 
_ of 1° in the value of A. We then require to know the length 
of the arc p’p’’, and we may, without sensible error, con- 
sider it to be described by the radius of curvature r = Po 
for the point P, through an angle p’op’’". Now 


tld , tt A’ , AS vt 
pop. — Xop — op eee a (5 +") - 


A’ Ne 
>i ul a 


SIMPLE CURVES 93 - 


By eq. (80) 


: We : ’ 
o = 1 — 2u’) and o” = (1 — Qu’ 


I 


and since ¢’ is so nearly equal to ¢’’ we may assume u’ 
/ er tr 
“=u; hence ¢’ — ¢” = pee 5 . 
(A’ — A’) (1 — x). 
But the condition of the problem requires A’ — A” = jis 
hence p’op’’ = (1 — u)°, 


U 


(1 — 2u) and p’op”’ 





Hig. 35. Fig. 36. 


Therefore the length of are p’p”’ for a change of 1° in the 
value of A is 
ky = r(l — wu) X are 1° 


or (Table XX) i = r(1 — u).0174533 
and since r = vL (Table VII,.2), 
L = v0 — wL .0174533 (83) 


By this formula Table VII, 3 has been prepared, for various — 
values of A and L. 

150. Given: two curves of the same length L but of 
different radi, starting from the same point of curve in a 
common tangent, to determine the direction and length of 
a line joining their extremities. Fig. 36. 

Let AX be the common tangent, and AP’, AP” the two 
curves, to determine the direction and length of P’P”. 

If we take the point P on the are P’P”’ determined by the 


/ Mt 
angle A = x — 





and draw a tangent PK to the valvoid 


. 94 FIELD ENGINEERING 


at P, we may assume without material error that the chord 
P’P” will be parallel to PK for any value of P’P”’ not exceed- 


ing +L, a limit not likely to be exceeded in practice. 
Let O be the center of the curve AP fixing the point P; 


/ } 
then AOP =i, and - 
OPK = peo 
2 
PRO ='K = oon. A TO) 


Since P’P” is assumed parallel to PK, 


a my + A’ 


REP OY GO = AS Ke 5 (Ll — 4) 


5 ae A”(1 ae u’’) NG (1 ean u’’) 


P’P”O” = 5 (84) 
Similarly producing P’’P’ to any point H, 
PPO) ees ee see (85) 
whence also 
eile Mille ay Gea a (85a) 


The slight error involved in the above assumption is cor- 
rected by taking out the value of « (Table VII, 1) correspond- 
ing to A’’, the less of the two given central angles; we have 
therefore written wu with the double accent in eqs. (84) 
and (85). 

When 7’ and 2” are positive, they will be deflected as in 
Fig. 36, on the side of the radius farthest from A; should 7” 
~ be negative it will of course be deflected from P’’O” toward A. 

The arc P’P” corresponds to a change of the central angle 
from A’ to A’’; hence 


Te A eer 


or 

PYP™ = (Al — AM (86) 

in which l; is taken from Table VII, 3, for L = AP, and 
x Of iN, + DNA 
= Sareea 


As in practice, the distance P’P” is usually small com- 


SIMPLE CURVES 95 


pared with Z, the are and chord will be almost identical 
and no further calculation is necessary. If P’P’’ is large, 
it will be found that eq. (86) gives the length of arc very 


I / 


A 5 2 
correctly when ——5—— does not exceed 20°, and the length 


/ ? 


of chord when ——s exceeds 60°; for intermediate mean 


angles it gives a value to P’P”’ between that of the are and 
chord. The arc P’P’’ may be considered to be described 
SPeee = (Table 
VII, 2), and its total curvature is found by multiplying 
its length by the degree of curve corresponding tor 
(Table I). 

Exam ple.—Given, a 2° 30’ curve, and a 1° curve of 12 sta- 
tions each from the same P.C., to determine the distance 
between their extremities. 


by the radius r = vL, v being taken for 


? ih) 
Aye ote X 19-80%. Ae 199 3 ~ =21°, 
Atos AM Ls 18% ul’ = 33446 
Eq. (84). i” = 2°,9737 = 2° 58/ 25” 


Kq. (85a). 0’ = 1” + A’ X A” = 20°.97387 = 20° 58’ 25” 
Eq. (86). Are P’P” = 18° X 10.425 187.65 ft. Ans. 


Eq. (82). r = 1200 X .7479 = 897.48 ft. = (say) a 6° 23’ 
curve. 
Total curvature, P’P’’ = 6°.383 X 1.8765 = 11°.9777. 


(The distance P’P’’ may be found by solving the triangle 
formed by itself and the long chords of the curve AP’, AP”’.) 

151. Given: a curve AP, to find a curve starting from the 
same point A, that shall shift the station P any desired dis- 
tance PP’ to the right or left. Fig. 36. 

Before we can determine what distance PP’ is desired, 
we must know (approximately) its direction. We have 
given, therefore, D, L, and A to find the angle OPP’, and 
(after measuring PP’) to find A’ and D’. 

The solution is necessarily somewhat approximate, yet 
close enough for all practical purposes. For if the required 
value of D’ were obtained precisely, it would probably in- 


96 FIELD ENGINEERING 


volve some seconds, and would therefore be discarded in 
favor of some value in even minutes. 
When P’ is inside the given curve: 


Eq. (80), «=OPK =wuA. Table VII, 1. 





Eq. (82). pe Po. = wl. Table Vig: 
Let 6 (delta) = degree of curve corresponding to r, by 
Table I. 
yereseton “RE Les Wok 
OPP! =1— 100" -36 nearly. 
Bq. (86). eH A+". Table VIT3. 
I 
Instead of taking l; from Table VII, 3, for the exact value 
of A it is well to take it for the estimated value of A : = 
Ba, (20) pr = 100 


When P’ is outside of the given curve: 


~=uA, r= ov, 


; rig oe XE 
180° — OPP’ =i-+ 100 -56 nearly. 
A gE, 2, 


Example——Given, a 4° curve of 800 feet, or A = 32°, to 
find a curve from the same P.C. which shall shift the last 
station, in, about 55 feet. (Fig. 36.) 


t= 32° X .3355 = 10°.736 





r = 800 °° 17450 = 596, .. 6 =.9° 86’ = 0°.6 
OPP’ = 10°.736 — 5D, x 4°.8 = 8° 06’ 
100 ‘ 
, Bo, Ors. ane 
Albee + ger = 40 
py oe cat = 5°. Ans 


SIMPLE CURVES Q7 


For a 5° curve, the true distance PP’ = 55.53 
BA 59" te Y* “fo PP = 'b4:60 


which proves this method practically correct. 

152. Given: a tangent and curve, and a straight lin®é 
intersecting them, making a given angle with the tangent at 
a given point, to determine the distance on the line from 
the tangent to the curve. Fig. 37. 


de 





HEN E Sie 


We have OA, AG, and the angle AGP to find GP. 


R 
‘ OG . sin PGO 
sin OPI = fi sin PGO = sin AGO 





sin (OPI — PGO) 
sin PGO 


When AGP = AGO, eq. (24), 
GP = R ex sec (90° — AGO) 
When AGP = 90°, §§ (92), (119), 
GP = R vers POA, sin POA = 
When AGP’> AGO, we have 
P’GO = AGP’ — AGO 


but the other formulas remain unchanged. 


PG=R 


GA 
k 


98 FIELD ENGINEERING 


Example.—Let R = 955.37, AG = 350, AGP = 40°. 

Ans.» AGO = 69° 52.47", OPI = 32°02’ 36” and'GP = 

72.40. 
« JLhis problem may be used in passing from a tangent to a 
curve when the tangent point is obstructed. The distance 
AP on the curve is defined by the angle AOP, which is readily 
found. 

If AGP’ > 2AGO the line will not cut the curve. 

Approximate Method. The point P may be located by 
‘‘ string intersection’? which amounts to placing two stakes 
on the line GI (one each side of P) and 
finding by trial, the point where the 
curve cuts a string connecting the > 
two stakes. 

Remark. If G@ is considered the V 
of the curve and the distance GP and 
angle AGP are measured, the radius 

_of a curve which will pass through the 
point P may be found by first solving 
the equation for angle OPI and then 
the one which follows for R. The 
problem will then be a modification of 
that in § 144. 

153. Given: a curve and a distant 
point to find a tangent that shall 
pass through the point. Fig. 38. 

We have the curve adg and the 
point P visible, but distance unknown, 
to find the point of tangent B. 

First Method. Any chord, as Of, 
parallel to the required tangent, if 
produced will pass the point P ata 
perpendicular distance equal to the 
middle-ordinate of that chord. Rang- 
ing across every two consecutive 
stakes on the curve we at first find 
the range falling outside of the required tangent, as 

bcG, cdH, etc.; but finally the range falls inside, as deKk. 
We then know that the required point is between c and e. 
If the range ce falls inside the point P, a perpendicular 
distance equal to the middle-ordinate of ce, the tangent 





Fie. 38. 


SIMPLE CURVES 99 


point is at d. If the perpendicular distance is greater 
than. this, the point B is between c and d. If less, or if 
the range ce falls outside of P, the point B is between 
d and e. The middle-ordinate for ce (200 feet) equals the 
tangent offset for 100 feet, given in Table I, and it is generally 
so small that it can be estimated at P without going to lay 
it off. 

To find the exact point B, when it falls between d and e, 
find by trial a point x (not shown) on the are cd in range with 
e and a point inside of P a perpendicular distance equal 
to the middle-ordinate of ex. The point B is at the middle 
point of the arc ex. If the point B is between c and d, stand 
at c and find a point x on the arc de in the same way. B 
is at one half the arc cz. 

The middle-ordinate of any chord ez is less than M for 
200 feet, and greater than m for 100 feet. If necessary, 
its exact value m’ can be found by 


, _m XK ex? 


= “16,000 So) 


m 
and this equation is nearly true when ez is as great as 300 
or 400 feet. That is, middle-ordinates on the same curve 
are to each other as the squares of their chords very nearly. 

By this method the point B is found without the use of 
the transit, so that the plug can be driven at B before the 
transit is brought up from the rear. It is therefore prefer- 
able to the following solution. 

Second Method. Fig. 39. From any two points a and c 
of the curve measure the angles to the point P, so that with 
the chord ac as a base, and the measured angles, we may 
find cP by the formula 


sin caP 
sin cPa 





Cts 


Knowing the angle c that cP makes with a tangent at c, 
we find the length of the chord cd by cd = 2R sinc. 
By Geom. Table XLI, 24, 
PB = Pe = VcP X dP 


whence we know ce. Opposite e, or on the arc eB described 
with the radius Pe, we find B. 


100 FIELD ENGINEERING 


154. Given: two eurves exterior to each other, to find 
the tangent points of a line tangent to both and its length 
between tangent points. Fig. 40. 

Let B and A be the required tangent points. Let OB = R, 
and O'A = Rh’. 

On the curve of greater radius R select a point H supposed 
to be near the unknown tangent point B, and knowing the 
direction of the radius OH, find on the other curve a point 
K having a radius O’K parallel to OH, and measure HK. 





Fig. 39. Fie. 40. 


In the diagram draw Ob and O’a perpendicular to HK, 
_ Then the angle KO’a = 90° — HKO’ = KO’A nearly, 
which is the angle required. We have therefore to find 
the correction aO’A = x, and apply it to KO’a. 

Aa =-R’ vers KO'a; Bb = R vers KO’a nearly. 

Ka = R’ sin KO’a; Hb = FR sin KO’a 

Bb — Aa = (R — R’) vers KO'a 

ab= HK + (Rk — R’) sin KO’a 
(R — R’) vers KO’a 

HK + (Rk — R’) sin KO’a 
KO'A = (KO’a — x) = HOB 


sin x = 





nearly. (88) 


SIMPLE CURVES 101 


Observe that KO’a = the angle between the tangent at K 
or H and the line HK; and KO’A = the angle between the 
tangent at K or H and the required tangent BA. 

If, instead of H and K, the points H’ and K’ had been 
selected, then 





sin 2 = —— ig) Mee nearly, (88a) 


H'K’ — (R — R’) sin H’Ob 
H'OB = K'0'A = H’0b +x. 


and 





Fia. 41. 


The length of BA should be obtained by measurement, 
but it may be calculated by 


AB = ab — (R — R’) sina (89) 


When R = R’, « = 0, and HK is parallel to BA. 
In case the curves are reverse to each other, as in Fig. 41, 


(R+R’) vers KO’a 
HK+(R+R’) sin KO’a 


KO’A = HOB = KO'a—- «x 
If the points H’ and K’ are selected, Fig. 41, 


(R + R’) vers H’Ob 
H’K’ — (+ RB’) sin H’0b 


H'OB’= K'O'A = H'Ob +z. 


Silt = nearly, (90) 





sin © = nearly. (91) 


102 FIELD ENGINEERING 


The lines HK, AB, and OO’ all intersect in a common. 
point J, Fig. 41. 
WH KEKE 








BS pap ee 
IB = VHI(HI + 2R sin HOb) (93) 
AB = p= +% (94) 


These last three equations furnish another method of 
solving the same problem. They may be applied to Fig. 40. 
by changing the sign of R’, 





In Fig. 41,if R = R’, then H] = 4HK and AB = 27B. 
155. Given: two curves, O and O’, reverse to each 
other, joined by a tangent BA’, and terminating in another 
tangent, B’F; to change the position of the Point of 
Tangent B of the first curve, so that the second curve may 
terminate in a given parallel tangent, B’F’. Fig. 42. 
Let X be the required new position of B; 
O” be the corresponding position of O’; 
NG — A’O’'B’ and NG mA St) ip 


Since the radii and the connecting tangent are unchanged 
in length, and all rotate together about O as a center, O” will 
be on a circle passing through O’, described with a radius 
OO’, and the required angle BOX = O’00”. 

In the diagram, produce O’A’ and. draw the perpendicular 
OG, and let a = the angle OO’G. Also, draw OK parallel 


SIMPLE CURVES 103 


and O’’K and O’H perpendicular to B’/O’. In the triangle 
OO’G we have 





cot OO'G = Oe or cota = eae (95) 
and a 
go” a aks (96) 
COS a 
The angle KOO’ =O00'BR’ =a+ 4’. 
The angle KOO” = OO”B"” =a + A”. 
KO = 00" -cos (a2 + A”), HO =O0O' cos (a+ A’). 
HK = O00’ {cos (a + A’) — cos (a + A’)] = BF’ 
cos (a + A’’) = cos (a + A’) + aa (97) 
BOX = O'00” = (a+ A’) — (a+ A”) (98) 


If we conceive a line to be drawn through O bisecting the 
are O’O"’, the angle it makes with B’’O” is a mean between 
B’0’O and B”’O"0; hence the chord O’O”, perpendicular 
to this line, makes an angle with O’P perpendicular to B’O’ 
of 

PO'O” = ila + 4!) + (@ +a") 


and since 


HT 


O'P = PO” cot PO'O” 
F’B" = B’F’ cot (a + A’) + (a+ A") (99) 


which gives the distance, measured on the parallel tangent, 
between the old tangent point and the new. 

This problem occurs in practice when both the connecting 
tangent and the radius of the last curve are at their minimum 
limit, and the parallel tangent is inside of the old one, as in 
the figure. Should the new tangent be outside, the same 
formulas apply, only changing the sign of B’F’ in eq. (97). 
But in this last case it is usually preferable to employ prob- 
lem § 136 or § 137. 

Example.—A 1° 40’ curve is followed by a tangent of 200 
feet, and that by a 4° curve of 10 stations ending in a tan- 
gent; and the offset to the given parallel tangent is 80 feet 
on the inside. Required, the position of the new tangent 
points X and B”’. 


104 FIELD ENGINEERING 


Here R = 3487.87, R’ = 1432.69, BA’ = 200, B’F’ = 80. 


Eq. (95) R + R’ 4870.56 log 3.687579 
BA ->"200, log 2.301030 

ats 2° 217 log cot 1.386549 
Eq. (96) a 2° 21 log cos 9.999635 
as OO' 3.687944 
Eq. (97) B’F’ 80 1.903090 
01641 8.215146 


a + A’ 42° 21’ cos .73904 
a+ A” 40°56’ cos .75545 


Eq. (98) BOX 1°25’ «. BX =85 ft. Ans. 
Eq. (99) PO’O” 41°38’ 30” cot 1.12468 X 80 = 89.97 = F’B’ 





156. When the tangents of a proposed road are to be in 
general much longer than the curves, it is desirable to estab- 
lish the tangents first in making the location, and afterwards 
determine suitable curves. On the other hand, if the curves 
necessarily predominate, they should be first selected and 
adjusted to the ground with reference to grade and easy 
alinement, and afterwards joined by tangents. In the latter 
case the field work cannot be successfully accomplished 
unless the location has been previously worked out upon a 
correct map constructed from the preliminary surveys. 
The map should show contours of the surface, and also the 
grade contour, or intersection of the surface and plane of 
the grade. In side-hill work the grade- contour indicates 
approximately the degree and position of the necessary 
curves. In the work of selecting proper curves upon the 
map, templets or pattern curves are almost indispensable. 

Pattern curves are furnished by dealers in sets of twenty, 
representing as many curves in series ranging from 1° up 
and cut to a scale of 50 feet to the inch, or of 100 feet to the 
inch; also in series ranging from 0° 30’ up, cut to a scale 
of 400 feet to the inch. These apply fairly well to maps 
drawn to other scales by making proper correction. Thus, 
the 4° templet of 100 feet to the inch may be used as a 2° 
curve upon a map drawn 200 feet to the inch, or as a 1° 
curve upon a map of 400 feet to the inch, 


SIMPLE CURVES 105 


Curves of 50 minutes and multiples of 50 minutes are most 
convenient in the field for making the necessary deflections 
to plus points between stations by reason of the simple 
ratio of minutes to feet; but such templets are not furnished 
by the trade. They may be cut in the office from Bristol 
board by means of a sharp knife point set in one socket of 
a beam compass. Scale and degree should be marked on 
each. 


TABLE OF CONVENIENT CURVES 

















Ratio of Min. Ratio of Min. Ratio of Min. 
D. to Feet. D. - to Feet. D. to Feet. 
50’ leee2 40’ PE 30! 3:10 
1° 40’ leek 12520 ues 1° 00’ Bp ats 
2° 30’ Bye eS 2° 00’ 6:5 1° 30’ 9:10 
3° 20’ PA | 2° 40’ Si5 can: 635 
4° 10’ Eore 32720" Qatk Domo Bre 
5° O” Sauk 4° 00’ iy BaF 3° 00 9:5 
5° 50’ ee 4° 40’ Tae 3° 30 21: 10 
6° 40’ tye], pray 1655 4° 00’ 1235 
7° 30° 9:2 6° 00’ 1e5 4° 30’ 27:10 
Bomeu: aa 6° 40’ va 5° 00’ 
9° 10’ Wear ee 120; eee ay 8104 oo 710 
10° 00’ 631 8° 00’ Peas 6° 00’ Sarno 











Paper Location. After the curves and tangents have been 
blocked out upon the preliminary map in pencil the line 
should be stationed with stepping dividers set precisely 
to the scale of the map (which may have shrunk slightly), 
testing them on a stretch of, say twenty stations. The 
angles between tangent lines are now determined, tangent 
distances computed and laid off to scale, fixing each P.C. 
and P.T., recording the plus on the map, and the bearing of 
the tangent. Notes for use in the field are taken from the 
map with memoranda of crossing points and prominent 
objects by which to correct in the field any discrepancies 
that may arise. (See § 54.) If the work has been well 
done these notes may be followed in the field with scarcely 
any alterations, and the tangents need not be extended to 
their intersections in many instances. But with a poor or 
incomplete map the line to be located may be only roughly 
indicated on the map, leaving the actual location to be 
worked out om the ground at:a greatly increased expense. 


LUG = FIELD ENGINEERING 


of time, not to mention the necessity for further revision 
in some places. 

In difficult country a trial profile may be read off from the 
contours of a good map, and certain revisions of the pencil 
line may be made before going into the field with a locating 
party. This precaution is of great importance when the 
line is to be laid to the maximum limit of grade, 


CHAPTER VI 
COMPOUND CURVES 


A. Theory 


157. A compound curve consists of two or more consecu- 
tive circular arcs of different radii, having their centers on 
the same side of the curve; but any two consecutive arcs 
must have a common tangent at their meeting point, or their 
radii at this point must coincide in direction. The meeting 
’ point is called the point of compound curve, or P.C.C. 
Compound curves are employed to bring the line of the road 
upon more favorable ground than could be done by the use 
of any simple curve. 

When a compound curve of two arcs connects two tangent 
lines, the tangent points are at unequal distances from the 
intersection or vertex, the shorter distance being on the line 
which is tangent to the are of shorter radius. 

158. Let VA, VB (Fig. 43) be any two right lines inter- 
secting at V, and A be the deflection angle between them. 
Let A and B be the tangent points of a compound curve 
(VA less than VB), and let AP, PB be the two arcs of the 
curve. The center O, of the arc AP will be found on AS, 
drawn perpendicular to VA; the center O2 of the are PB 
will be found on BS produced perpendicular to VB; and 
the angle ASB will evidently equal A. Join VS, and on 
VS as a diameter describe a circle; it will pass through the 
points A and B, since the angles VAS, VBS are right angles 
in a semicirele. Draw the chord VQ, bisecting the angle 
AVB, and join AQ, BQ. Then AQ, BQ are equal, since 
they are chords subtending the equal angles AVQ, BVQ. 
From Q as a center, and with radius QA, describe a circle; 
it will cut the tangent lines at A and B, and also at two 
other points G and Y, such that VG = VA, and VY = VB. 
Hence BG = AY, and the parallel chords AG, BY are 


» 107 


108 FIELD ENGINEERING 


perpendicular to VQ. Join. AB; then AQB = ASB = A, 
since both angles are subtended by the same chord AB. 

In the triangle VAB, the sum of the angles at A and B is 
equal to the exterior angle A between the tangents; while 
their difference (A — B) is equal to the angle at the center Q 
subtended by the chord BG, which is the difference of the 





Fre. 48. 


sides (VB — VA). For the angle VAB = VAG + GAB, 
and the angle VBA = VBY — ABY. But VAG = VBY 
and GAB = ABY, and by subtraction VAB — VBA = 
2GAB = GQB, since A is on the circumference and Q at the 
center. 

159. 'THrorEmM.—The circle YAGB, whose center is Q, is 
the locus of the point of compound curve P, whatever be the 
relative lengths of the arcs AP, PB composing the curve. 


COMPOUND CURVES 109 


On the circle YAGB, and between A and G, take any point 
P, and on AS find a center O,, from which a circular are may 
be drawn cutting the circle at A and P; also on BS produced 
find a center Oo, from which a circular are may be drawn 
cutting the circle at B and P. Join PQ, PO, and PO. 
Since when two circles intersect, the angles are equal be- 
tween radii drawn to the points of intersection, @PO,; = QAO, 
and QPO, = QBO;. Draw the chord QS and it subtends 
the equal. angles QAO, = QBO,. Hence QPO, = QPOs; 
and the radius PO; coincides in direction with the radius 
PO», which is the condition essential to a compound curve. 

Now, if we imagine another point P’ to be taken on QP 
or on YP produced, and the ares AP’ BP’, drawn from cen- 
ters found on AS and BS, it is evident that the equality 
of angles found in respect to P could not exist in respect 
to P’. Hence the ares would intersect in P’ at some angle 
O,PO, and would not form a compound curve. Therefore, 
Q. E. D. 

160. THEorEM.—In any compound curve the radial lines 
passing through the three tangent points A, P, and B are all 
tangent to a circle having the point Q for its center, and for its 
diameter the difference of the sides VB and VA. 

Draw the three lines QN, QL, QM perpendicular to the 
radial lines BO2, POs, and AS respectively. Then the three 
right-angled triangles BQN, PQL, and AQM are equal, since 
BQ = PQ = AQ = radius of the circle AGB, and the angles 
at B, P, and A are equal by the last theorem. Hence 
QM = QL =QN, and if a circle be described with this 
radius about Q, the three lines BO2, PO2, and AO; produced 
will be tangent to it. Draw QI/ perpendicular to VB; it 
will bisect the chord GB in I; andQN = BI = 3BG. Hence 
the diameter 2QN = BG = VB — VA; which was to be 
proved. 

Corollary 1. The compound curve intersects the circle 
AGB in the point P, at an angle equal to half the difference 
of the angles VAB, VBA. For QPL = QBN = BQI = 
2BQG. The are AP is exterior, and the arc PB interior to 
the circle AGB. 

Cor. 2. Since both centers are on the line PL, the position 
of the point P fixes the lengths of the radii of a compound 
curve. As P is moved toward G both radii are increased, 


110 FIELD ENGINEERING 


until when P reaches G, AO, becomes AK, a maximum, 
while BO, becomes infinite. As P moves toward A both’ 
radii are diminished, but the least value of the are AP depends 
upon the least radius allowed on the road. If in the diagram 
we make AO, equal to the least radius allowed, a right line 
drawn through the point O, tangent to the circle LMN 
fixes the corresponding minimum value of the arc AP, and 
also of the radius BO, for given values of VA, VB, and A. 
Between these limits any desired values of the radii may 
be employed. 

Cor. 3. In the triangle SO,O2, the sum of the two central 
angles AO;P and PO>2B is equal to the exterior angle ASB =A; 
consequently, as the central angle of one arc is increased 
by any change in the position of the point P, the central 
angle of the other will be diminished an equal amount. 

Cor. 4. Only one value of the angle AO,P is consistent 
with a given value of the radius AQ,, since both depend 
on the variable position of the line PL; and for the same 
reason only one value of the angle BO.P is consistent with 
a given value of the radius BO.. Hence only one radius 
or one central angle can be assumed at pleasure, the remaining 
garts being deducible therefrom in terms of the sides VA, VB, 
and the angle A. 


B. General Equations 


161. Let S, = the side VA = the shorter tangent; 
S. = the side VB = the longer tangent; 
R, = the radius AO; 
R, = the radius BO; 
y = diff. VAB — VBA; 
A =the sum VAB+ VBA; 
Ai = central angle AO,P; 
A, = central angle BO2P. 





In the triangle BQ/, cot BQI = oe But JQ = VI X 
cot JQV = 3(S2 + S:) cot $A, and BI = 4(S8, — §,). 
Se +81 
ses 1 
cot 37 ea cot 3A (100) 


By Cor. 3, Ait A,=A (101) 


COMPOUND CURVES LE 


In the triangle AQM, AO, = AM — MO,. But AM = 
MQ cot #y, and MO, = MQ cot 3A\. 
Now MQ = 3(S2 — S81) 


Ry = 3(S2 — Si)(cot dy — cot $A1) (102) 
Similarly, Re = 3(S2 — Si)(cot av + cot ZA2) 
Subtracting, 
Ry — kh, = 3(Se2 — S,)(cot 4A2 + cot 4Aj) (103) 
R, 
i = oS aia era ee a 
cot 7A; = cot 7 SES 
From (102), * (104) 
ye Se A re ee a 
cot zAe Wiceeakey cot av 
In the triangle ABG, 
: AB sin BAG 
BOE® sin AGV 
or vee 
1 - eee SIDA y 
3(S2 — S1) ans (105) 


by which we find $(S. — Si), when, instead of the sides and 
A, we have given AB, and the angles VAB and VBA. 
From (103), 


MS: — 8) = (106) 
cot 37 = Tes + cot 3Ai 
From (102), (107) 
cot 47 = Woe — cot 4A, 
From (100), 
Ty Sete 8) = 3(Se — S81) cot oy (108) 


cot. 3A 


S, and S; are found by adding and subtracting the values 
obtained by eqs. (106), (108). 

From (105), 
(Se — S:) sin 3A 


JAB = : 
, sin 3 


(109) 


which may be used instead of (108) when the sides are not 
required. VAB’= 3(A+ 7) and VBA = 3(A — 4). 


112 FIELD ENGINEERING 


162. Given: the sides VA = 8S, and VB = 8S. and the 
angle A; assuming the shorter radius h,, to find A:, A:, 
and Ro. 

Use eqs. (100), (104,) (101), (102), and (18). 

Example—Let VA = 1899.90, VB = 1091.12, A = 74°, 
and assume Ry = 955.37. 














(100) 2(S2 + Si) 1495.51 log 3.174789 
2(S2 — S81) 404.39 ‘* 2.606800 
‘* 0.567989 
2A 37° cot ‘‘ 0.122886 
ie 2Y 11° 381’ 01.5 cot 4.90769 ‘* ‘* 0.690875 
(104) R,(D = 6°) ‘¢ 2.980170 
1607 ey) ‘* 2.606800 
2.36249 ‘“ 0.373370 
a zA1 212 24! cot 2.54520 
(101) 3A 37° 
oi 7A2 15° 33’ ‘* 3.59370 
(102) 37 ‘ 4.90769 
8.50139 ‘* 0.929490 
LS eet Si) ‘* 2.606800 
R2(D = 1° 40’) 3.536290 


(18) . A, = 42°54’, ly = 715; As = 31°06’, Le = 1866. 


163. Given: the line AB, and the angles VAB, VBA; 
assuming the longer radius Ro, to find Ao, Ai, and Ry. 

Example—Let AB = 2437.82, VAB = 48°31’, VBA = 
25° 29’, and assume Ry = 3437.87. 

Use eqs. (105), (104), (101), and (102). 

Ans. 4(Se2 = S)) ae 404.38; 5A = 15° 33/5 Ay = 21° al 4 
and log R; = 2.980192 with D = 6°. 

164. Usually a compound curve is fitted by trial to the 
shape of the ground, after which it may be desirable to 
calculate the sides VA, VB, or the line AB, and the angles 
VAB, VBA. 

Example.—From the point of curve A, a 6° curve is run 
715 feet to the P.C.C.; thence a 1° 40’ curve is run 1866 feet 


COMPOUND CURVES 113 


to the P.T. Required, the sides VA, VB, and the line AB, 
and angles VAB, VBA. Here Ry = 955.37, Ai = 42° 54’. 
Ry 403487.87, Ag = 31° 06’. 

Use eqs. (106), (107), (108), and (109). 

Ans.» S2 = 1899.90; Si) ='1091.12'; VAB =.48°31’; 
VBA v=925%29' ‘and AB. —"2437,82. 

165. Given the radii Ii, Ro, the angle A, and one side, 
VA, or VB, to find the other side and the central angles A, 

In the triangle AMQ, AO, = AM — MO, = I1Q — MQ 
cot MO,Q; or 


Ri = 4(S2 + S81) cot 4A — 4682 — 8) cot FA; 
whence 
2(S2 + Si) = 3(S2 — Si) cot ZA; tan 3A + RF, tan 3A 
By eq. (106), 


sin 4A, sin 4A, 
sin 3A 





2 (Se - Si) = (Re — Ri) 


Substituting this above, subtracting and reducing 
sin $(A — Aj) 
4gin A 


Si = (Re = R1) sin 3Ae ok Rk, tan ZA 


But 3(A._— Ai) = 3A2 and 2 sin? 4A, = vers As, whence . 


_ (Re — Ri) vers A, + Ri vers A 








a sin A 
Transposing, 
_ 8, sin A — R, vers A 
vers Ao = FA RR 7 OP (111) 
Similarly, from the triangle BQO,, 
Re = 3 (Se + S1) cot 2A + 4(Se —— Sy) cot 5 Ae 
from which and eq. (106) we derive 
cial R, vers A us (R. — FR) vers A; (112) 
sin A 
and april 
we reiathee fe vers A — S2 sin A (113) 


Lige—- Lbs 


Example.—Given: VA = S; = 1091.12, A = 74°, and the 
radii R; = 955.37, Re = 3487.87, to find Ai, Ao, and S2. 


114 FIELD ENGINEERING 


Use eqs. (111), and (112). 

Ans. Ag = 31° 06’; Ai = 42° 54’; S, = 1899.90. 

166. Given one side, and the radius and central angle of 
the adjacent arc, to find the other radius and side. 

From eqs. (111), (113) we have 


S; sin A — R, vers A 
Rz £4 Ri ice 1 1 
vers Ag 





(114) 
Pein R, vers A — So sin A 
vers Aj; 


by one of which the required radius may be found; the 
required side is then found by eq. (110) or (112), as in the 
last problem. 





Hie. 44; 


Example.— Given: VA = SS; = 1091.12, A = 74°, Ry = 
955.37 and A; = 42° 54’; to find Ry. A, = 74° — 42° 54’ = 
31-06". 

Use eq. (114) and (112). 

Ans. R,—R,=2482.52; R, = 3437.89. 

Otherwise: Fig. 44. If convenient in the field, a tangent 
PV. may be run from the point P to intersect the farther 
tangent. The distance PV2 multiplied by cot 3A. will equal 
the radius 2 by eq. (25). 

167. Remarks.—If the first are AP be produced to G, 
Fig. 44, so that AO,G = A, then G is the tangent point of a 
tangent parallel to VB, and by § 137, the tangent point B 


COMPOUND CURVES 115 


must be on the line PG produced. Conversely, if the point 
B is assumed, and the arc AG given, the point P must be 
on the line BG produced. The radius R; may be found by 
BP 

Soria ai LAY 
similar triangles R, : R, :: BP : GP. 

The distance VD, Fig. 48, from the vertex to the circle 
AGB is expressed by the formula 


BP being measured on the ground; or by 





VD = 8, cos $ (tan > ~tan4 er ") (115) 
If the point P falls at D, then VD is also the distance of 
the curve from the vertex measured on the line VQ. But 
when P falls at D, the radius PO, is perpendicular to the line 
- AB, and Ai = VAB, and A, = VBA. When A; is greater 
than VAB, the arc AP, being exterior to the circle, cuts the 
line VD; but when A, is less than VAB, the are PB cuts 
the line DQ. 
If the line O:P produced passes through V, we have 





sin 3A (116) 
1 ; 


giving A; = 4A + QVL and A; = 3A — QVL. 
When A, is greater than this, we have for the external 
_ distance of the vertex 


HE, = R,; ex sec AOiV 


in which the angle AO,V is found by the formula cot AO,V = 
“ and H, is measured on a line VO,, making the angle 
AVO, = 90° et AOwV. 

When 4, is less than ($A + QVL), we have similar expres- 
sions with respect to the arc BP and center Oz. 

168. To locate a compound curve when the point of com- 
pound curve is inaccessible. Fig. 45. 

Each are being in itself a simple curve is located as such. 
When the P.C.C. is accessible, the transit is placed over it, 
and the direction of the common tangent found, from which 
the second are is then located. 

When the P.C.C. is not accessible, the common tangent 
ViV2 may be found by locating the points V; and V2, which 


116 FIELD ENGINEERING 


may be easily done, since ViA = ViP = R; tan 34;, and ~ 
Vo2B = V2P = R. tan $Ae2, from which each are may then 
be located by offsets or otherwise, as in the case of simple 
curves. 

Should the points ViV2 be obstructed, the common tangent 
may be found by an offset HG = LP from any convenient 
point H, for knowing the angle HO,P, we have HG = R; 
vers HO,P, and GP = R, sin HO,P. 

If the entire tangent ViV2 is too much obstructed for use, 
the parallel line HK may be employed, observing that the 
angle POsK is found by vers PO.K = ae 
LK by LK = R, sin PO2K, by which a point K on the second 
are is found having a tangent offset KAJ = HG. 





, and the distance 





Fia 45. Fia. 46. 


Should the line HK be also obstructed, we may run the 
inverted curve HP’ = HP and P’K = PK to find the point 
K from which so much of the second are as is accessible 
may be located. 


C. Special Problems in Compound Curves 


169. Given: a compound curve ending in a tangent; to 
change the P.C.C. so that the curve may end in a given 
parallel tangent. Fig. 46. 

Let APB be the given curve ending in VB; 

V'B’ be the given parallel tangent; 
p = perpendicular distance between tangents. 


It is required to change the point P, and with it the values 


COMPOUND CURVES ays 


of A; and As, so that with the same radii R; and R, the new 
curve AP’B’ may end in the parallel tangent V’B’. . 

a. When the tangent V’B’ is inside of VB: 

Let Ay = AQiP, Ay’ = AOTE- Ag = PO-2B, thet = P’0,'B’, 
and in the diagram draw OiG perpendicular to BO,; then 
GO2z = 0,02 cos As, KO,’ = 0,02’ cos As’. Subtracting, 
since O02 == O,0,' = (Re = R,), and KO,’ == GO, = GB — 
1G aie 

C p = (R2 — R,)(cos Ae’ — cos Ae) 
whence 
COS Ad’ = oe + cos Ag (117) 
PO,P’ = (Ag — Ao’) and the point P is advanced. 
b. When the tangent V’B’ is outside of VB: 


fo) = (Re = Fi) (eos Ag — cos Ao’) 
whence 
cos Ae’ = cos A, — ee (118) 
POP’ => (Ad! a Ao) and the 
point P is moved back and the 
arc AP diminished. 

In case the curve termin- 
ates with the arc of shorter 
radius, or k, follows Re. 
Fig. 47. 

c. When V’B’ is inside of 

VB: 





Fig, 47. 
p = (R, — R1)(cos A: — cos Ai’) 


whence 


pe MS eS 
cos Ai’ = cos Ay TR, (119) 
PO.P’ = (A;’ — Ay) and the point P is moved back. 
d. When V’B’ is outside of VB: 
p = (Rz — R1)(cos Ai’ — cos Aj) 
whence 


pae Pp 
cos A;’ = cos A; + oe Bi (120) 


s4 


PO.P’ = (A; — Ai’) and the point P is advanced. 


118 FIELD ENGINEERING 


Example.—Let R = 2292.01, Ri = 1482.69, As = 28°, 
and p = 20.07 inside of VB; case a. 





Pp 20.07 log 1.302547 
(117) fe — Ry 859.32 fe 2.934155 
.023356 ‘‘ 8:368392 
Ao 28° cos .88295 
A’s 25° ‘« 906306 
PO,P’ 3° 


170. Given: a compound curve terminating in a tangent, 
to change the P.C.C. and also the last radius, so that the 





Fie, 48: 


curve shall end in a parallel tangent at a point on the 
same radial line as before. Fig. 48. 

Let APB be the given curve ending in the tangent VB; 
let V’B’ be the given parallel tangent; and let p = BB’ = 
HI = the perpendicular distance between tangents. 

It is required to change the point P to P’, and also the 
value of Rz to R2’, so that the new curve may end in V’B’ 
at B’ inside of VB on the same radial line BO». 

In the diagram produce the arc AP to G to meet O,G 
drawn parallel to O2.2B; then PO,G = As. Draw the chord 
PB, and it will pass through G. Lay off the distance p 
from B on BO, to find B’; draw B’G and produce it to inter- 
sect the arc APG in P’, Then P’ is the P.C.C. required. 
Join P’O, and produce it to meet BO: produced in O2’. Then 


COMPOUND CURVES 119 
P’0,/ = B’O,’ = R,’, the new radius, with which describe 
the are P’B’. 

By Geom. Table XLI, 18: 

PBV = }P0;B = }A2, and GB’V’ = }P‘O,'B’ = 3A.’ 


PGP’ = BGB’ = (A, — Ay’) 





Fia. 49. 


Draw O.X perpendicular to BOs. 
Then OK = b'H = bl = O02 sin Ao = (Re = Ry) sin Ao. 


tan 4A, = os tan 4A,! = GER GI sa? 








Puen 
RS WS © LIA oe ie 
tan 7As vig 7A Cee fay aa (121) 


In the triangle 0,002’ 
sin Ao’ : sin Ag :: 0,02 : O,02’ :: (Re — Ri) : (R.’ == Ri) 


ras at sin Ao at 
ee Fe a ae 





and 





ALP sin A 
Ra! = (Ry — Ry) — a; + Ry (122) 


120 FIELD ENGINEERING 


If B’V’ were outside of VB, 





Lavine Wig: BABI Lad Eas 
tan 3A,’ = tan 3A, 4+- (ET tee aan (123) 
Ae % sin As } 
R,’ = (Ro — Ff) ea + Rk, (122) 


When the smaller radius R, follows R2: If: the given 
tangent B’V’ is inside of BV. Fig. 49. 

















eal i sil Pp 
tan 3A, tan A; + ee Pans (124) 
jh a x Ray sin 7 
Ry’ = Ry — (R, — R,) iad (125) 
If B’V’ is outside of BY: 
daer ye Be ey: 
tan 9Ai tan 3A; (Recut A; (126) 
ir. es ie sin Ay 
Ri’! = Re — (Ro — R,) a Ait (125) 
Example 1.—Fig. 48. 
Let Ry = 2292.01 p = 20.07 inside. 
FR, = 1482.69 A, = 28°. 
(121) KR. — fs = 859.32 log 2.934155 
Ag 282 log sin 9.671609 
2.605764 
p 20.07 1.302547 
.04975 8.696783 
tan 5A» 24933 
F. tan 3A,’ . 19958 vRNA LD) Bis 
(122) Ad! 22° 34’ sin 9.584058 
(R2 — R,) 2.934155 
; 3.350097 
As 28° sin 9.671609 
(Ro — Ri) 1051.25 3.021706 
Ry, 1432.69 
Ans. PR,’ = 2483.94 «. D = 2° 18’ 25” 


PO,P = 28° — 22°34’ = 5°26’. PP’ = 135,83 ft, 


COMPOUND CURVES 121 


Example 2.—Fig. 49. 
Let R, = 2292.01 p = 20.07 inside. 
Ri = 1482.69 Ai = 46° 


Ams. Fy’ = 147441. «)) .D = 3°53’ 12” 


3°.1166 
_ 2.5 


Observe that in either figure both tangents must be on the 
same side of the point G, in order to a solution. 

171. Given: a compound curve ending in a tangent, to 
change the last radius and also the position of the P.C.C., 
so that the curve may end in the Same tangent. Fig. 50. 





PO2P’ = Ay’ = 41 = 3°07’ .. are PP’ = = 124.67 ft. 





Fia. 50. 


I. When the curve ends with the greater radius Ro. 

Let APB be the compound curve in which fi, Re, A; and 
A, are known. 

In the diagram draw the chord PB and produce the first 
arc AP to meet it in G; draw O,G, and produce it to meet 
the tangent in K. Then by § 1387 O,K is parallel to O2B, 
by eq. (57) 

GK = (Re res Ry) vers Az (127) 


If we assume P’ as the new P.C.C., we have Ao’ = P'O.’B", 
and the chord P’G produced will intersect the tangent at 
the new point of tangent B’, and B’O,’ = R,’. Similar to 
eq. (127) we have 


GK = (R,’ — FR) vers A,’ 


122 FIELD ENGINEERING 


and equating the two expressions, we obtain 
(R. — R,) vers As GK 


fame ero vers Ao’ rag vers Ay’ es 
If we assume R,’, we have 
, Re — Ri pet ee 
vers A;’ = Roh vers Ay = Rr ee (129) 


In the two right-angled triangles BKG and B’KG, we have 
BK = GK cot 4A: 
B’K = GK cot 4A)’ 





Fiq. 51. 


and by subtraction, 
BB' = GK (cot 4A2’ — cot 4A2) (180) 


in which GK is obtained from eq. (127). 

When BB’ as given by eq. (130) is negative, the point B’ 
falls between B and V. 

If we assume the distance BB’ on the tangent, we have 
from the last equation, | 

cot 4A’ = cot 4A, + EY (181) 
GK 

GK being obtained from eq. (127) and R2’ from eq. (128). 
In eq. (131) use the + sign when B’ is beyond B as in 
Fig. 50, * 


COMPOUND CURVES 123 


It. When the given curve ends with the smaller radius 
fae Higebl: 
We have by a similar reasoning 





CEE (Re = R,) vers A; (182) 
pe a (R. — R) vers A; oa GK 
pHa les Perales in pre at 8) 
OTRAS Foe oN GK 
. Nia == 
vers Ai" BE Bet vers A; = ee Pa (134) 
= GK (cot 4A; — cot 4A;’) (135) 
BB’ 


cot sAy’ = cot ZA) Le (136) 


. GK 
using the — sign when B’ is beyond B. 





Hre.so2; 


Example.—Fig. 51. 

Let R. = 2291.01, R, = 1432.69, A, = 46°, and let the: 
P.C.C. be moved back 200 feet from P to P’; hence PO,’ P’ = 
5° and A,’ = 51°; to find the new radius R,’ and the dis- 
tance BB’. 

Use eqs. (132), (133) and (135). 

Ans. FR,’ = 1584.16; D = 3° 37’; and BB’ = 68.04. 

172. Given: a compound curve ending in a tangent, the last 
radius being the greater, to change the last radius and 
also: the position of the P.C.C. so that the curve may end at 
the same tangent point, but with a given difference in the 
direction of ‘the tangent. Fig. 52. 


124 FIELD ENGINEERING 


Let APB be the given compound curve, PO; = R; and 
PO, = Rs a Ri. 

Let V’B be the new tangent, and the angle VBV’ = i, 
the given difference in direction: to find BO,’ = Ry! , BOP’ = 
A,’ and the angle PO,P’. 

We have 


BO, — 0,0. = R, — (Re. — Ry) = Ri 
BO,! — O,0,' = Ry’ — (Ro’ — Ri) = Ri 


From which we see that whatever may be the value of the 
new radius, the difference of the distances from B and O; 
to the new center is constant, and equal to Ry. We therefore 
conclude that the centers O, and O,’ are on an hyperbola of 
which B and OQ, are the foci, and R,; the major axis. 

This suggests an easy graphical method of solving the 
problem. 

Through B draw a line perpendicular to the new tangent 
V’B which will give the direction of the required center O2’. 
On this line lay off BK equal to R;, and since (R,’ — Ri) = 
0,0,’ = KO,’, if we join KO, the triangle KO,’O, is isosceles; 
therefore bisect KO, and erect a perpendicular from the 
middle point to intersect the line BK produced in Oy’. 
Draw O,’0; and produce it to intersect the are AP (produced 
if necessary) in P’. Then P’ is the new P.C.C. required, 
and BO,’ = P’O.! = Ry)’, the new radius. 

The analytical solution is as follows: 

Adopting the usual notation of the hyperbola 


Let 2a = R, = the major axis; 
2c = BO, = the distance between foci. 


Produce the are AP and through B draw the tangent 
BH, and join HO; = R,. Then in the right-angled triangle 
BHO, ; 

BH? = BO,? — R,? = 4c? — 4a? 


Now by Anal. Geom., c? — a? = 6?. 

Therefore 2b = BH = the minor axis. 

Draw the chord PB and produce the are AP to cut it in G. 
Then by Geom. (Table XLI, 24), ) 


BH? = PB X GB = 2Ry sin 3A; X 2(Rz — R,) sin 445 


COMPOUND CURVES 125 








Bi = 2 sin LAV R; (R, sae R,) (137) 
Let a = the angle HO,B, then 
BH Ry 
t = — = 
ana and BO; aha (138) 
In the triangle BO,O2 let O:BO. = 8; then 
: 2—-h, . 
sin B = “BO, * sin Ap (139) 


The polar equation of the hyperbola for the branch [0,0.’, 
taking the pole at B and estimating the variable angle v 
from the line BO,, is 

62 
- ¢-cosv —a 

When v = B +1, r = R2’, and substituting the values of 

a, b, and c found above, we have 


BH? 


~ 2(BO, cos (8 + 4) — Ri) (140) 





: R,! 


using (8 +7) when V’ falls between V and A, as in the 
figure, and (8 — 7) when V’ falls beyond V. 
In the triangle BO,O,’, the angle BO,’0; = A:’ and 


; bert BO, : ; 
sin: As = Risk, sin (8 +7) (141) 
Finally 
PO,P! = Ag — (Ae! +7) (142) 


Remark.—When V’ falls between V and A, as in Fig. 52, 
if the angle 7 be greater than the angle VBH, the curve 
ceases to be a compound, and becomes reversed. There- 
fore VBH =a —&8 is the maximum value of 7 possible 
in this case. When V’ falls beyond V, the point P’ will 
fall between P and A; and the largest possible value of 
7 will then be that which renders PO,P’ = Aj, and makes 
the point P’ coincide with A. 

Example.—Fig. 52. Let Ri = 1432.69 A; = 31° 

4 60 cnn Rep =2292.01 . As = 56° 


Use eqs. (137), (138), (139), (140), (141) and (142). 


126 _ FIELD ENGINEERING 


Ans. log BH} 3:119825%s icc: =42.% 36’, 23"",7; 
log BO’ =3.289262; B = 21° 28’ 06.3; Re’ = 2949.05; 
Ae’ = 36° 18’ 26”; PO,P’ = 18° 41’ 34""= 342.3 feet. 
Remark.—This problem may also be solved by first finding 
the new sides V’A, V’B, from which and the new central 
angle (A +7), and the radius Ri, may be found Aj’, A2’, and 
R,', as in § 162. The new sides are readily found from the 
old ones by solving the triangle VBV’. If the original 
sides are not given, they must be calculated as in § 164. 
173. Given: a compound curve ending in a tangent, the 
last radius being the less, to change the last radius and the 





Fie 53. 


position of the P.C.C. so that the curve may end at the same 
tangent point, but with a given difference in the direc- 
tion of tangent. Fig. 53. 

Let APB be the given curve, and PO,= R:, and 
PO, = Ri < Re. Let V’Bbe the new tangent, and VBV’ =7 
the given angle; to find BO,’ = R,’, BO,'P’, = A;’, and 
the angle PO.P’. 

We have 


BO, + O,02 = Ri + (R, = hi) = Rk, 
BOY’ +; O71'02 oa Ry’ ae (Re a Ry’) — Re 


from which we infer that the locus of the center O,’ is an 
ellipse, of which B and O:2 are the foci, and R, the major 
axis, since the swum of the distances BO,’ and O.0,’ is always 
equal to Ro. 


COMPOUND CURVES HAN 


This suggests an easy graphical solution of the prob- 
lem, as follows: 

Perpendicular to V’B draw the indefinite line BK, which 
will contain the required center O,’, and lay off BK = Ro. 
Join KO:, bisect it, and from the middle point erect a per- 
pendicular to intersect BK in O;’. Join O.0,', and produce 
the line to intersect the arc AP (produced if necessary) 
in P’, which is the new P.C.C. required. P’O,'’= BO,/= R,’ 
the required radius, and P’O,'’B = Aj’. 

The analytical solution is as follows: Adopting the 
usual notation of the ellipse, 


let 2a = R, = the major axis, 
2c = BO, = the distance between foci. 


~ At B erect BH perpendicular to BO; to intersect the are AP 
(produced if necessary) in H, and join HO, = R:. Then 


BH? = R,? — BO? = 4a? — 4c? 
But by Anal. Geom., a? — c? = 6. 
Hence 2b = BH = the minor axis. 


In the triangle BO,O2 we know BO; = Ri, and 0,0, = 
Rz — R:, and the included angle BO,O, = 180° — Ai; hence 
_ by Trig. (Table XLII, 25) 

2k, — Rp 


tan 4(O,0.B ae O:.BO2) = : R — tan 4A, (143) 


The angles at B and O; are then found by Table XLII, 
26). Let B = the angle O.B0:2; then 





sin Aj 





BO, = (Re ae Rk) sin B (144) 
The value of BH? above,may be written 
BH? = (R2 + BO) (Re. — BO2) (145) 


The polar equation of the ellipse, taking the pole at B, 
and estimating the variable angle v from the axis BO:, is 
b2 
r= 
@ — c-COS v 


128 FIELD ENGINEERING 


When v = 8 ¥ i, then r = Ry’, and substituting the values 
of a, b, and c, given above, we have 


BH? 


Ry ~ 3(R, = BOs cos (6 Hay) 


(146) 
using (8 — 7) when V’ falls between V and A, as in Fig. 53, 
and (8 + 7) when V’ falls beyond V, 

In the triangle BO,’O2, the angle O,’BO, = (6 + 7), and 
the exterior angle BO,’P’ = A’; hence 


sin Ay’ = wos sin (B 1) (147) 
pple 1 
Finally PO2P! = (Ai = 1) — Ay’ (148) 


When VY’ is on AV, then POP’ is negative, showing that 
it must be laid off from P toward A; but when V’ is beyond 
V, then POP’ is positive, and P’ will be on AP produced. 
The only limits imposed on the angle 7 are that the resulting 
value of PP’ shall not exceed PA, and that FR,’ shall not be 
less than a practical minimum. 

Example.—Fig. 53. 


Let Dz, = 3° 20’ Ry =1719.12 A, = 23° 20! 
D, = 6° Rh, = 955.37 A, = 48° a= 7° 45/ 


lI 


The resulting values are as follows: 


6B 21° 09’ 32/..6 
BO, 1572.42 log 3.196567 
BH? ‘* 5.683829 
Ry’ 1273.65 ** 3.105052 
Ay’ 54° 56’ 
PO2P’ . 14° 41’ 
nee 440.5 


(See also remark at end of § 172.) 
174. Given a simple curve joining two tangents, to replace 


it by a three-centered compound curve between the Same 
tangent points. Fig, 54. 


COMPOUND CURVES — 129 


Let R = AO = radius of simple curve. 
Ry = POr= P'O; < BR Ay = PO,P' 
R, = AO, = BO;> R A, = AO.P = BO3P’ 
A = AOB 
Since AQ, is made equal to BO; and VA = VB, AO.P must 
equal BO;P’, and the compound curve will be symmetrical 
about the bisecting line VO; and the center O; will be on the 


line VO. 
We have at once from the figure, 


2h2 + Ay =A (149) 





In the triangle 00,0, we have 


0:0. : OO, :: sin AOV : sin PO,V 
whence 
(R, — R) sin dA 


Bae) Bey sin $Ai 


(150) 
which expresses the general relation between the quantities, 
R and A being given. 
We may now assume values for Ry and Ry subject to the 
above conditions, viz., Ri < Rand Rh, > R; whence 
} ’  (R, — R) sin 4A 
TAc pe a) 2 
sin 3A ey (151) 
In selecting values for R, and Rs, the degree of curve D,; 
should be but little greater than D of the simple curve, say 
from 30 to 60 minutes, while Dz, may be taken at 3D to 7D. 
_ Example——Given: & =171912 D=3°20' 4 = 40° 


130 FIELD ENGINEERING 


Let R, = 1482.69 D, = 4° 
R, = 5729.65 D,=1° 





R.—R 4010.53 log 3.603202 
R,—R, 4296.96 ‘< 3.633161 
‘* 9.970041 
3A 20° log sin 9.534052 
2Ai 18236"-57" ‘¢ 9.504093 

Ai Bl ike. oA, 

Ao 1222) 937 


AP = P’B 138.4 feet 
Again we may assume Az and R,, whence 
and 
ree Rsin $A — R, sin 4A; 


; : 2 
sin $A — sin 3A; (152) 


Example.—Given: R = 1719.12 A = 40° 
Let R, = 1432.69 Ao = ic soe Lage 38° 
Ans. Re= 738724 9. °D, = 0° 463’ AP = 129: 


Finally we may assume A and R2, and deduce A; and R, 
from eqs. (149) (150); but this is the least desirable because 
the value of R; so found will not usually give a convenient 
value to the degree of curve Dj. 

175. To determine the distance HH’ between the middle 
points of a simple curve and a three-centered compound curve 
joining the same tangent points AB. Fig. 54. 

In the triangle 00,02, we have 


sin Ao 
sin 3A 
HH’ = O00, + 0,\H’ — OH 


OO; ae (Re — R,) 





sin Ao 
sin 4A 


HH’ = (Ry — Ry) 





— (Rk — R,) (153) 


In the first example given above HH’ = 14.55, and in the 
second HH’ = 17.05 feet. 

In many instances the distance HH’ is so great as to render 
this problem practically useless, unless the distance HH, is 


COMPOUND CURVES ical 


discounted beforehand by putting the simple curve AHB a 
sufficient distance inside of the proper location through the 
point H’. But the problem given below is usually prefer- 
able. 

176. Given, a Simple curve joining two tangents to re- 
place it by a three-centered compound curve which shall 
pass through the same middle point H. 


I. The curve flattened at the tangents. Fig. 56, 





Prey 5D: 


Let R = AO, the radius, and A = the central angle of the 
simple curve AHB, and let H be the middle point. 
Let Ry => PO, = HO, Ai = POP" 
Rz = PO, = A’O. = B’O3 A, = PO,A’ = P’03B' 


and, A’ and B’ be the new tangent points required. 


We have at once, as in the last problem, 


2A, + Ai =A (154) 


Since the curve is to be symmetrical about VO, HP = HP’. 
PA = P’'B, and AA’ = BB’. 

In the Haeean produce the are HP to G, and draw O.G 
parallel to OA, and produce it to K. ‘Then a tangent line at 
G will be piiralicl to VA; and by § 1387 the point G will be 
on the long cherd HA, cal on the long chord PA’. GK is 
the perpendicular distance between parallel tangents, and 


132 FIELD ENGINEERING 
the problem is similar to that given in $171; whence by 
eq. (57) we have, in this case, 


GK = (f, — Ri) vers As = (A — Ri) vers 3A (155) 


for the general equation in which RF and A are given. 
Analagous to eq. (130) we have 


AA! = KA’ — KA = GK cot GA'K — GK cot GAK. 
AA’ = GK (cot $42 — cot 4A) (156) 


in which GK is obtained from (155). 
We may now assume values for Ry and Ro, making Ry < Rk 
and R, > R, and deduce the values of As, Ai, and AA’. 
Solving eq. (155) 
|) Ce si vers: ais oh GIe 
vers Az = BU ee, Sys es 





(157) 


Eq. (154) gives A;, and eq. (156) gives AA’. 
Example.—Fig. 55. 


Given: R= 764.489 D aio A = 40° 
Let Ry 716.779 D, = 8° 
R, = 3437.870 Dz; = 1° 40’ 
Use eqs. (155) and (156) and (157). 
Ans. A2= (say) 2° 38’, AA’ = 108.87. 


I 


Again, we may assume Az and Rk; < R; whence 





Ai = A — 2A¢ 
and 
eq. (155) GK = (R — R,) vers 4A 
and ae 
Ry a Ri =| vers Ao (158) 


Eq. (156) gives AA’. 
Again, we may assume Az and the distance AA'; whence, 
from eq. (156) 


AA’ 

OS speobRApsuleotghs ye 
GK 
2 


eq. (158) gives Re. 


COMPOUND CURVES 133 


Again, we may assume Rk, < R and AA’; then, eq. (155) 
GK = (R — R,) vers 4A 


and eq. (156) 
AA’ 


cot 3A. = cot 4A + Gk (160) 
and eq. (158) gives Re. 
Example.—Vig. 55. 
Given: R = 764.489 D=7°30' A=40° 
Let Ry = 716.779 De 5 


AA’ = 110 
Use eqs. (160) and (158). 
Ans. R, = 3476.3; D2 = 1°39’; AP’ = 157.0, A; = 
104° '46'. 


II. The curve Sharpened at the tangents. Fig. 56. 
This case will occur only when, with a given external 
distance VH, a simple curve would absorb too much of the 
tangents. 
Let AHB be the simple curve, and 
A’'PHP’B’, the required compound curve 


R, = PO, = HO;; Ae = PO2P’ 

hy. = POj=.A'0, =,B’03; A, = A’O,;P = P’0;B’. 
We have from the figure, | 
Ba 4A, = A (161) 


In the diagram draw O2G parallel to OA, cutting the tan- 
gent at A, and produce the are HP to G. Draw the chords 
GH and GP, passing through A and A’ respectively. We 
have then a discussion similar to the preceding case, and to 
the problem § 171, Fig. 51, whence we derive the general 
formulas: 

GK = (R. — R:) vers Ai = (Rp — R) vers $A (162) 
and 
AA’ — GK (cot 4A; — cot +A) (163) 


1. Assuming Ri < RF andk, > R 


GK R, —R 
vers Ai = nea pe R Ty vers 3A (164) 





134 FIELD ENGINEERING 


2. Assuming’A; < 4A and hi < & 


BR vers 34 —_Ry vers Ax 


Re 
vers 34 — vers Ai} 


3. Assuming A; < $4 and AA’ 





AA’ 
Ch cot 4A; — cot 1A 
Gk 
fas fare vers 3A 
GK 
Ry ae ica 
vers Aj 


4, Assuming R, > R and AA’ 
GK = (Ry — R) vers 3A 


, 
cot $A; = cot tA + ee 





GK 





(165) 


(166) 


(167) 


(168) 


(169) 


The third assumption will usually secure most readily the 
desired curve. AA’ should be assumed as small as the nature 
of the case will allow, and A; should not be much smaller 


than 3A. 


It is evidently not necessary that the new curve should 
be symmetrical; for having laid out the curve A’PH, the 
simple curve HB may then be used, or, if desirable, some 
compound curve HP’b’ determined by an assumed value 


of BB’ not equal to AA’, 


COMPOUND CURVES 135 


These formulas (154) to (169) are readily adapted to the 
case of substituting a compound for a simple curve when it 
is necessary to keep one tangent point fixed, but to move the 
other a certain distance in either direction on the tangent. 
For if in Figs. 55, 56, we draw a tangent at H, and make H 
the fixed point of tangent, it is evident that the central 
angle of the cvrve will then be AOH. The only change 
necessary, therefore, to adapt the formulas to this case is 
to write A in place of 4A, and to observe, instead of eqs. 
(154), (161), that 


Ay ++ Ao =A. 
Example.—Fig. 55. 
Let R = 1910.08 A = 84° 
Assume AA’ = 260. A; = 38° oe es 


Use eqs. (166), (167) and (168). 
Ans.) Rp -= 5294.15:~R, = 1193.88; D = 4°48’; A’P = 
791.67; .and PH, = 369.23: 


177. Given, two curves joined by a common tangent 
to replace the tangent by a curve compounded with 
the given curves. Fig. 57. 


Let FR, = BO; the radius of one curve; 
R; = AOs the radius of the other curve, > 1; 
1 = BA the common tangent; 
R, = POs = P’O, the radius of connecting curve; 
A, = POP’ the central angle of connecting curve; 
a = AO3P’ and 6 = BO;P; 
1 — AO;0,. 


In the diagram join 0,0; and draw O,G parallel to BA. 
Then in the right-angled triangle O:GO; we have, 





G0, Rs — Ri 

cot i = Go, = 5 (170) 

0,0; = ie Pa = pl. (171) 
COs 7 sin 2 


which gives the distance between the centers of the given 
curves. 

We shall now assume the following geometrical truths, 
which may be éasily demonstrated. 

If two circles intersect in one point, they intersect in two 


136 FIELD ENGINEERING 


points; and the line joining the two points is the common 
chord. 

The common chord is perpendicular to the line joining 
the centers, and when produced it bisects the common 
tangents. 

If a third circle is drawn touching the two circles, a tangent 
to the third circle, parallel to the common tangent, will have 
its tangent point on the common chord produced. 

Conversely, therefore, if the tangent BA be bisected at K, 
and a line, KJ, drawn perpendicular to 0,03, KJ will coincide 
with the common chord produced, and the angle [KA = 





Fic. 57. 


AO;0, = 7. If on KI we assume a point J through which 
it is desirable that the connecting curve should pass, then 
I is the tangent point of a tangent parallel to BA; con- 
sequently a line through J perpendicular to BA contains 
the required center Oy. 

I. Let p = HI = the perpendicular distance between the 
tangents. 

If in the diagram we join JA and JB, and produce the 
chords to intersect the given curves in P and P’, then P and 
P’ are the points of compound curvature; and the lines PO; 
and P’O; produced will intersect JO, in the same point O.; 
and the angles P’O.J = a, and PO.I = p. 

In the triangle AJB the line KJ bisects the base AB, and 
we have by Geom. Table XLI, 25, 


COMPOUND CURVES 137 


I 


Af?+ BP = 2AK? + 2KT? 
By eq. (56) AI = 2(R, — R3) sin fa 
Bl= 2(Re = FR) sin 48 
AK =i and KI =—2. 
sin 7 
; 1 Ay a eat 
4(Re — R3)*? sin? 3a + 4(R, — R,)? sin? $6 = $12 + — 


sin? 2 





Dividing by 2 and putting vers a = 2 sin? ia and vers 
B = 2 sin? 46 (Table XLII, 46), 


(Roan) vero ee — Ri? vers B = 30 + —P, 
: sin? 7 

But by: eq. (57) : 
(R, — Rs) versa = (R, — R,) vers B = p (172) 


p(2he — (R3 + R,)) = 42? + Pe 











sin? 2 
Diy a (Pecan eel 173 
mer Tee : 4p” sin? 7 G73) 
From (172) 
: eet 0 ae ee? 
_ vers a jaa a vers 6 aa Bi (174) 
“ and from the figure 

Az =a+B (175) 


These formulas solve the problem when p is assumed. If 
desirable we may find a and 6 independently of Re, for in 
the triangle AJB, IAB = 4a and IBA = 38; and since 
HK = p cotz2, 


Aliecrst = Bie \~-k 


cot ja = a ae aE: = ay — cot2 (176) 
BH i1+HkK l : 
aL =_l oO BENE ean = 
cot 3 ial : Dp + cot 4 (177) 


II. In case a or B is assumed, we have from the last 
equation 
l l 


2(cot fa + cot i) 2(cot 38 — cot 2) ge) 


Da 


138 FIELD ENGINEERING 


Ill. In case the radius Rz is assumed, then in the triangle 
O,0:0; we know all three sides; for OO. = (Re — Ri), 


O03 = (Re — Rs), and 0,03 — Rs = He 





By Trig. (Table XLII, 31) 


vers Ag = 0,0. X 00; 


in which s = 4 sum of the three sides. 
Substituting values, and reducing, observing that, 


( -1) ( 2 =) = sec? 1 — 1 = tan?2 


COS 2 COS 2 





and that (R; — R,) tan7z = 1, we have 
[2 
2(Re Las Ry) (Re acd Rs) 





vers Ag = 


(179) 


In the same triangle 


MO. 0A— sin = pt eee 
0.0, 


for from the figure 0;0,0. = 7 — 6, and taking the value 

of 0,0; from eq. (171) 

(Re, — Rs) sin A» sin 7 
l 


We then find a from eq. (175) and p from (172). 
The angles a and 6 may be found otherwise, for by Trig. 
(Table XLII, 27) we have in the triangle 0,020; 


0,03 =: 0.03 





(180) 


sin (1 — B) = 














sin $(0,0302 — 030102) = 0,0, cos Ae 
ve : fs ‘ — R; — R,) cos 7% cos $A 
sin (90 - (i AS 5 *)) aos 
cos ( + “54) = cos 2-cos ZA (181) 
which is a convenient formula when 7 and Ae are not too 
small. Having obtained = ss B , we have 
a=s+55" pain -25" (189) 


2 


COMPOUND CURVES SD 


_ For a constant value of 1 the less the difference of R; — R; 
the greater will be the value of the angle 7. When R; = R, 
cot 7 = 0 and 7 = 90° and the tangent point J will be on a 
perpendicular to BA drawn through the middle point K; 
and a = 6. On the contrary, as (R; — R,) increases, 7 
becomes less, and the foot, H, of the perpendicular HJ 
moves toward Bb, the tangent point of the curve of smaller 
radius R;. The distance HK = pcoti. The connecting 
curve is farthest from the tangent BA at J. To find the 
ordinate from BA to the curve at any other point, subtract 
from p the tangent offset for the length of curve from J to 
the ordinate in question. §115, eq. (39) may be used on 
flat curves with tolerable accuracy, even when the distance 
equals several hundred feet. 

IV. It is evident that in this problem R. must be greater 
than either R, or R;. As the center Os is taken nearer the 
line 0,03, R, grows less, and is a minimum when Q, falls on 
the line 0,03. In this case we have A, = 180°, and 


R, = 3(Rs + Ri + 0103); a minimum. (183) 


This limit must be regarded in assuming the value of Re. 
Since 


0,02 _ O03 = (Re =e Ri) — (Re = Rs) = (Rs = Ri) 


a constant value, independent of R2, we infer that the center 
O2 is always on a hyperbola of which O; and O, are the foci; 
(R; — R,) equals the diameter on the axis joining the foci; 
and 1 equals the diameter at right angles to it, for in the 
‘triangle OiGOs, 


2 = 0,0, — (hs — Ri? (184) 


Exam ple.—Fig. 57. : 
Given: R, = 1432.69 R; = 1910.08 and / = 400. 
Assume p = 11.4 to find Ro, a and B. 
Use eqs. (170), (173) and (174). 
Ans. i = 39° 57° 34”; Rs; = 3489.59, called 3437.87; 
a = 7°00’; B = 6° 07’ nearly. A, = 14 07s. 
Example.—Fig. 57. 
Given: R:,= 1432.69, R; = 1910.08, and 1 = 400. 
Assume fR, = 3437.87, to find Ao, 8, a and p. 


* 


140 FIELD ENGINEERING 


Use eqs. (179), (170), (180), (175) and (172). 

Ans. Aa = 13° 07' 22"; i= 39° 57.34"; B= 6° 06’ 55’; 
a = 7°00’ 27”; and p = 11.41, 

178. Given: a three-centered compound curve to replace 
the middle are by an arc of different radius. 

I. When the radius of the middle arc is the greatest. 
Fig. 57. 

First find the length and direction of the common tangent 
AB. Let Ad: = central angle of the middle arc, hz = its 





Fia. 58. 


radius, ‘and FR, and Rs the radii of the other arcs. From eq. 
(179) 





l = V2( Rs _ R,) (Re aa R3) vers Ao (185) 


Then find 7 by eq. (170), a and 6 by eqs. (179a), (175), and 
p by eq. (172). . 

For the new arc we may now assume a new value for 7, 
or for Ra, or fora. Indicating the new values by a prime, 
if we assume p’ we proceed as in the last problem, using 
eqs. (173), etc. If we assume R,’, we use eq. (179), ete. 
If we assume a’, we use eq. (178). 

Il. When the radius of the middle arc is the least of the 
three. Fig. 58. 

In this case the middle arc is within the other two pro- 
duced; and for the same values of Ri Rs and 0,03, the locus 


COMPOUND CURVES 141 


of the center O2 is the opposite branch of the hyperbola 
found in §177. When the center O: falls on the line 0,03, 
A, = 180°, and 


R, = 3(Rs + Ri — 0103), a maximum. (186) 
Analogous to eq. (185), we have 


bese VY 9 Fei =| ha) Ra ce Feo) Vers As (187) 


which gives the length of the common tangent YZ =GO,. 

We then have the values of i and of 0,0; by eqs. (170), 
(171), and of a and 8 by eqs. (181), (182), and analogous 
to eq.’ (172), 


p = (Rh, — R22) versa = (R3 — Re) vers B (188) 


‘in which =p is the perpendicular distance HI between parallel 
tangents. : 

For the new arc we may now assume a new value for p, 
for R2, or a. Indicating the new values by a prime, if 
we. assume p’, we have, analogous to eq. (173) 

















l? p! 
92 , — — 
2Re Rs + Ri (= ++ vat ; (189) 
and from eq. (188) 
ry es p’ ; / se p 
vers a’ = Fain vers 8 Paeay. . (190) 


If we assume R,.’, we have, analogous to eq. (179), 


12 
2(R, — Rp')(R3 — Ry’) 
and we find a and B by eqs. (181), (182), and p’ by eq. (188). 

Ill. When the radius of the middle arc has an intermedi- 


ate value, compared with the other radii. Fig. 59. 
In this case, whatever be the value of Re, we have 


0302 + O20; = (3 — Re) + (Re — Ri) = (Rs — Ry) 





vers Ay’ = (191) 


a constant value independent of 2; hence we infer that the 
locus of Oo is an ellipse, of which O, and Q3 are the foci, and 
(R3; — R,) equal to the transverse axis. 

Let 1 = QQ’ =.the conjugate axis, and let 7 = QO;,0; = 
QO,03. Then QO; = QO; a 3(Rs — RR). 


142 FIELD ENGINEERING 


Produce O0;Q to G, making QG = O;Q, and join GOr. 
Then by similar triangles GO; is perpendicular to 0,03, and 
GO, = 1; and in the right-angled triangle GO;0, 

(192) 


(193) 





Fia. 59.) 


Analogous to eqs. (185) and (187), we have 
| 1 = V2(R; — R2)(R; — Ri) vers A (194) 
which may also be derived from the triangles 0,0.0; and 


O:03Q. 
Let a= 020301, and B = O.0,0; 





Then 
A Ly 0,02 . i Ry ci Ri . a 
sina = 0.0; sin A, = i tan 2-sin Ae (195) 
From the figure 8 = A, —a@ (196) 


In the diagram produce the line 0,0, and it will intersect 
all the ares. At the points Z and Y, where it cuts the inner 
and outer arcs, draw tangent lines perpendicular to O;0,. 
Draw the radius O2/ parallel to O;0,, and the tangent line 
IL at I. 

Let g = ZY and p = ZL = HI 


COMPOUND CURVES 143 


Then by the theory of parallel tangents, § 137, the point 
J is on the chord PZ produced, and it is also on the chord 
P’Y; and we have 


p = ZL = (R2 — R,) vers B (197) 

q—p=LyY = (Rk; — R2) versa (198) 

and q equals the swm of these. But q = ZY is the shortest 
distance between the inner and outer arcs, and has a constant 
value independent of R2. If we assume R, = 3(R3 + R,) the 


center O2 will be at Q, anda = 6 = 7, and p = 3q. Making 
these substitutions above, 


q = (R3 — R,) vers 7 (199) 
Also, from the figure, 


Ee Yi =— OsY = OZ ra 0103, 
or 
q = R3 — R, — 0,03. (200) 


In the triangle ZY we have by Geom. Table XLI, 26, 


ZI? = 1Y? + ZY? — 2ZY(ZY — ZL) 
or 
ZY? —2ZY-ZL =1Y? — ZI? 
Now, 
ZI? = 4(R, — R,)? sin? $8 = 2(R2 — R,)? vers B 


TY? = 4(R; — R2)? sin? 3a = 2(R3; — Re) versa 
Hence 

ZI? =2(R,—Ri)p and IY? = 2(R3 — R2)(q — p) 
Substituting these values, and solving for p, we have 


p= q(R; — Re — 29) _ Wks — R, — 29) 





201 
k;—hi-—@ 0,03 ( ) 
Also a 
R, = (Rs — 3q) — p- 4 : (202) 
For any other value of 2, we have 
, 0,0 
Ro’ = (Rs — 49) — p Egil 
Hence ve 
R,! — R, = ——(p — p’) , (203) 


*¢ 


144 FIELD ENGINEERING 


which gives the change in R, for a given change in the value 
of p. 
Observe that as p diminishes FR, increases and vice versa. 
Having determined the value of R,’, we find p’ by sub- 
stituting R,’ for Rz in eq. CO and from eqs. (197), (198) 
we have 


vers p’ = dirs sails (204) 
, ag Aa foag 7 
vers a’ = 5 ay (205) 


and the change in the points of compound curvature is found 
by (8 — B’) and (a@’ — a). 

Remark.—When fk. = $(3 + Ri), As = 27, a minimum, 
and the long chord PP’ is perpendicular to 0,03. When R; 
is greater than this, a is greater than 8, and vice versa. _What- 
ever be the value of R2, the long chord PP’ always cuts the 
line 0,0; produced in the same point S, at a distance from 
Z of 


ZS = Ry vers 7; 
or from O; of OS = R, cos 1. 
This item will be found useful in solving the problem 
graphically. 
Example.— 
Let Ry = (81,84 Dy = 7 20. 
Re. = 1375.40 Dz = 4° 10’ Az = 48° 
R; = 1910.08 Dz = 3° 00° 
p—p' = 11.30 


Use eqs. (194), (192), (193), (195), (196), (203), (200), 
(201), (197) and (198). 

Ans. log l = 2.661123; i = 23° 57’ 55’’; 0,03 = 1030.98, 
log 3.013249; a = 25° 19’ 52’’: B = 22° 40’ 08”’; g = 97.26; 
R," = 1495.18, (say) 1494.95 for 3° 50’ curve; p’ = 34.57; 
Bp’ = 17° 55% a= 31° 54; ol — o = 60 134 PP 
DIS80-"B = pa aren Te PP’ = 64,71. 

The practical difficulty in changing the middle are of three 
centered curves lies in the difference of measurement that 
ensues. Thus, in the last problem, although the total central 
angle is the same, the new curve is 6.56 feet shorter than the 


COMPOUND CURVES 145 


original, making a fractional station at P’”’. If the change 
is made during the location, it is well to re-run the last 
are from P’”’ to the tangent following, so as to eliminate the 
fractional station from the curve. 

Instead of the solution given above we may obtain A,’ by 

, j2 . 
vers A)’ = 5(R, — Ry')(Ry — Ry) 
derived from eq. (194); and then find a’ by eq. (195). 

Graphical Solution.—On any well-drawn plan of the curves 
we may try various curve templets touching the first and 
third curves until we find a new middle curve to suit the 
required conditions. 

We then take the value of its radius R,’ from Table I, 
subtract from R; and R; and with the differences, from the 
centers O; and O,, draw short arcs to intersect, thus locating 
O2'.. We then join this point with O, and O; and produce 
the lines to intercept the given curves in P” and P’”’, 
Finally draw the long chord P’P’”’, which must pass through 
the point-S. The angles may then be scaled, but are better 
computed as before. 


CHAPTER, VII 
REVERSED CURVES 


179. A reversed curve is a combination of two simple 
curves of opposite curvature. There is a common tangent 
at their point of junction. The outer rails of the two arcs 
are on opposite sides of the track at the point of reverse 
curve (P.R.C.) and this calls for the elevation of both sides 
at the same time. As there is no opportunity of elevating 
‘the outer rail at the P.R.C., reversed curves should not be 
used where speed is desirable, as on main lines. They 
may be used to advantage in cross-overs and yard work, 
and are often imperative in the location of spur tracks 
to manufacturing or industrial plants. 

Reversed curves may be divided into two general classes, 
according to whether the extreme tangents are parallel or 
diverging. 

180. Given: the perpendicular distance p, between the 
parallel tangents, and the common radius of the reversed curve, 
to find the central angle of eacharc. Fig. 60. 

Let APB be the center line of the reversed curve, DG the 


center line between tracks, ; 
AC = BC. =ir, and. ACPiS BC P= ne 
Then 
AD = BG = 3p 
and 
vers ACP = ie 

Cc 

or : 
vers Ar = re (206) 


If p and Ar are given, by transposing eq. (206) there 
-results 





pel (207) 
vers Ar 
146 


REVERSED CURVES 147 


181. Given: the perpendicular distance p between the 
parallel tangents, and the radius and central angle of the first 
arc of the reversed curve, to find the radius of the Second are. 
Fig. 61. j 

Let-APB be the center line of the reversed curve, AC = 7, 
BC’ =r, HB = py:ACP =. BC'P, = A,. 

Draw DG through the P.R.C. and parallel to AH. 


Then 


HB = p = AD+GB = CP vers ACP + C’P vers BC'P 
or 
p = 7 vers Ar + f2 vers Ar 
from which 


2 Pp 
mtn = Cae (208) 





Fig, 60, 


If 7:1, 72 and p:p are given, A; may be found by transposing 
eq. (208), giving 
Pp 


vers A; = —— 
m1 +72 


(209) 
182. Given: the perpendicular distance p between parallel 
tangents, the chord distance d, between the P.C. and the 
P.T. and the radius, ¥1, of the first arc of the reversed curve, 
to find the radius, Ye, of the second arc. Fig. 62. 
Let APB be the center line of the reversed curve, 


AB =d, AC =n, BC’ =r 


148 - FIELD ENGINEERING 


It may be proved that the line AB passes through P, 
the point of reverse curve. By drawing a line C’L parallel 
to BA and erecting a perpendicular to it, a triangle KCL 
is formed which will be similar to the triangle ABH. 


Then ‘ 
COC LIAI AS spr 


But 
CL =n-+7r, LK = 3AB and BH = p 
e (ri site), Vad ad) 2-7 
or 
a? 
ntn= 5 (210) 





Fia. 62. 


If ri, r2 and p:p are given, d may be found from eq. (210). 


d = V2(n + r2)p (211) 
With equal radii, 7; = rz = r and eq. (210) becomes 
ia 4p (212) 


and eq. (211) becomes 
d =2Vrp (213) 


183. Given: the length, 1, of the common tangent of a 
reversed curve and the angles of intersection with the 
diverging tangents, to find the common radius of a reversed 
curve connecting the diverging tangents. Fig. 63. 


REVERSED CURVES 149 


Let ViV2 be the common tangent of length 1, AV; and BV, 
the diverging tangents, HViP = A,, and GV2P = As; the 
angles of intersection. 


Then 
ViV2 — V,P + V.P 
or 
1 =r tan 3A; +7 tan 4A, 
whence 


l 
T = 
tan 3A; + tan 3A: 





(214) 


Table III may be used in an approximate solution as 
follows: Find the tangent distances for a 1° curve with 





Fic. 63. Fia. 64. 


central angles of A, and A», designating them by 7; and 72. 
Then the desired degree of curve will be 


D = Att (approx.) 


184. Given: diverging tangents, a line, AB, joining them, 
and the angles at A and B, to find the common radius 
of a reversed curve extending from A to B. Fig. 64. 

Let 1 = AB, EAB and UBG = the given angles A and B. 

Draw CM and C’S perpendicular to, and CS parallel to 
AB. 


150 FIELD ENGINEERING 


Let ANG = CHB = _C'CS =H. 
Then 
Also 


C'S = CC’ sin H = 2r simi 
C'S =C’N + CM =rcosA-+rcos B. 


Equating, 2r sin H = rcos A +rcosB 


or 
sin H = (cos A + cos B) 
Now 
AB =t= AM’+NB+CS 
or 
Ll=rsnA-+rsinB+ 2rcos H 
and. | 


l 
"sin A + sin B + 2 cos H 





(215) 


The station of the point P may be found from the degree 
of curve and the central angle C which equals 90° + A — H. 

185. The solution of curve problems may often be sim- 
plified by measuring a certain distance, or an angle and a 
distance in the field. 

In the case of diverging tangents for reversed curves, an 
easier solution may sometimes be found by prolonging one 
are until it is parallel to the other, thus using eq. (208) 
with the distance p known. 

186. Given: the angle of intersection of two diverging 
tangents, the distance from the P.C. to V. and the unequal 
radii of a reversed curve, to find the central angles of the 
- two arcs and the distance from V to the P.T Fig. 65. 

Let the following be the given quantities: 

AV, the given distance, A the P.C. of the curve, and 
AVT =A. 

Designate ACP by Ai, PC’B by Ao, CA by 1, and C’B by ro. 

Continue the arc PA to the point H where it comes to a 
tangent parallel to V7’. 

Then ET is the perpendicular distance between the parallel 
tangents and if designated by p may be found from eq. (208). 


ET = p = (1 +72) vers ECP 


Again ET = EK + AS =n versA + AV sin A 


REVERSED CURVES 151 


Equating, and since ECP = BC’P = A, we get, 


(7: + re) vers As = 7, versA + AV sin A 


or 
rare tate r, versA+ AV sindA (216) 
Ye fs 
Ay — Ag a A 


The distance VB = VS + AK — TB 
or | 
VB = VA cosA +7 sin A — (r1 +72) sin A, (217) 





Fie. 65. Fic. 66. 


187. Given: the angle of intersection of two diverging 
tangents, the distance from the P.T to V, and the unequal 
radii of a reversed curve, to find the central angles of the 
two arcs and the distance from the P.C. to V._ Fig. 66. 

The figure is similar to that in the previous section except 
the are AE is omitted and the lines C’U and MB are drawn 
perpendicular to AV and BH perpendicular to C’U. 

A tangent to the are at the point U will be parallel to 
- VA and from eq. (208), NU = p = (rn + f2) vers Ai. 


152 FIELD ENGINEERING 


But 
NU = UH+ BM =nversA+BV sin A 
_ 2 vers A + BV sin A 
vers A; = Shi sate er eae (218) 
and 
As =A, +A 
Also 


VA=VM+BH+NA 
= VB cosA+ 7m sin A+ (m+ 72) sin A; (219) 


CHAPTER VIII 


TURNOUTS AND CROSSINGS 


A. Turnouts 


188. A turnout is a curved track leading from one track 
to another. The switch rails turn the train from one track 
to the other, and at the point where the outer rail of the 
turnout crosses the rail of the main track a frog is intro- 
duced which allows the flanges of the wheels to pass the 
‘main rail. At points opposite the frog, guard rails are 
placed, their object being to prevent the wheel flanges from 
bearing against the frog point and also to keep them from 
being turned to the wrong side of the point. 

The stub switch, Fig. 67, is a form which is not much used 
at the present. time. The switch rails are common to both 
tracks. One end of each rail is free and can be shifted from 
one track to the other as required. The free end of the 
rail is called the point or toe of switch. The tangent point 
at A is called the heel of switch, and the distance, AD, is 
_the length of switch. The switch rail should be several feet 
longer than AD or BG, and the excess be spiked down in 
the line of the main track back of A or B. Then if the 
switch rails are thrown over to meet the rails of the turnout 
they will be sprung into ares coinciding with those of the turn- 
out, provided that they are made of proper length. The 
distance DK through which the point of the rail moves is 
called the throw of the switch. It varies from 44 to 6 inches. 

The split switch, Fig. 68, is the form most commonly used 
at the present time. In this form, one rail of the main 
track and the inner rail of the turnout are continuous. The 
outer switch rail when set for the turnout makes an abrupt 
angle with the main track. Both switch rails are planed at 
one end to a wedge point 4 of an inch in thickness. To 
prevent the inner switch rail from projecting inside the gauge 
rail when set for the main track, the inner stock rail of the 

3 153 


154 FIELD ENGINEERING 


turnout is bent at a point slightly back of the switch point. 
Fig. 68 shows the switch set for the turnout. 

The switch rails are fastened together by two or more 
tie rods, the first rod being connected either to the switch 
stand, or to a rod extending from the interlocking plant. 
The switch rails are free to move over their entire lengths, 
but are fastened at the heel of switch, points J and K, and 
hinge about these points. The distance gauge to gauge at 
the heel of the switch should be enough to allow for an angle 





Fie: 67. Bie 63 


splice and the insertion of a spike. This distance is about 

+ inches. The lengths of switch rails recommended by . 
the American Railway Eng. Assoc. range from 11 to 33 
feet. 

189. The Frog. Two general types of frogs are used, 
the rigid frog and the spring frog. The parts of the rigid 
frog are shown in Fig. 69. In the rigid frog, the wing rails 
are both fastened securely to the main part of the frog, 
with sufficient clearance for the flanges of the wheel. The 
spring frog is used where the bulk of traffic is on the main 
track. The main flangeway is always open, but the other 
is closed to afford a continuous bearing for wheels on the 


TURNOUTS AND CROSSINGS 155 , 


main track. The movable wing is held against the point 
by a spring, but is pushed aside by the flange of cach wheel 
passing the frog on the turnout. Frogs may be designated 
by their angles, but it is customary to designate them by 
numbers expressing the ratio of the bisecting line FC of the 
tongue to the base line, ab, Fig. 70. Observe that F is at 
the intersection of the edges produced, and not at the blunt 


eS ee 


> 
3 
by 
2 
i=.) 
— 
3 
~ 
<= 





Theoretical Point 


Throat 





Fic. 70. 


Fig... 62, 


point of the tongue. The point of intersection of the two 
lines is called the theoretical point and the blunt is the actual 
point. The actual point is usually 3 inch thick. 

In the triangle aFC, 


— = cot 4aFb 


and if we let » = the number of the frog, and F = the frog | 
angle, then 


N= = 574 = 3 cot AF (220) 


. 156 FIELD ENGINEERING 


If ¢ denotes the thickness at the actual point, then the 
distance between the theoretical and actual points = nt. 

190. In computing a turnout, both switch rail and frog 
may be considered curved; the switch rail may be considered 
straight and the frog rail curved; or both switch rail and 
frog may be straight. 

191. Split Switch. In the split switch both the switch 
rails and the frog are straight. A circular curve called the 
lead curve joins the two and is tangent to them. 

Given: a straight track of gauge, g, length of switch rail l, 
frog angle F, and number n, distance from theoretical point 





ihaite’, efile 


to toe of frog f, distance gauge to gauge at heel of switch h, thick- 
ness of frog point't, and of switch point w. To find the lead 
from point of switch to the theoretical frog point and also to 
the actual frog point; the radius of the lead curve yr and the 
length of the rails between heel of switch and toe 
of frog. Fig. 71. 

Let AD and EJ be the switch rails, DK the distance gauge 
to gauge at heel of switch, h, and S the switch angle DAK. 

Draw the construction lines shown in the figure. 

h-—w 


l 
The angle V’MI = F, and V’MK = 3(F — S) 
KMI =F —3(F —8S) =}(F+S) 





sin S; = (221) 


TURNOUTS AND CROSSINGS 157 


Now 
KI =DJ —DK—-—-MN=g-h-—fsnF 


IM = KI cot KMI = (g —h —fsinF) cot 3(F +S) 


and 


Then the theoretical lead = L; = distance from the switch 
point to theoretical frog point = BF = BJ + 1M + NF or 


L,=l+(—h—fsin F) cot}(F +8) 4+feosF (222) 

and the actual lead = La is 
a=l+(qg—h-—f sin F) cot fF +8) +f cos F4+nt 
(223) 


One value of GH(= KI) has just been found. A second 
value is, 


GH = GC — HC = (r + 3g) cosS — (r + 3g) cos F. 
Equating these two values and solving for r 
g=h—jen kl g—h—fsnF 
cos S—cosF 2sin 4(F —S) sin i(F +S) 

; (224) 





(r+ 29) = 


On FB lay off the distance FP = FM =f. 
Then the distance between the heel of switch and toe of 
frog on main track = JP = BF — BJ — PF 


or 
J Pee igh Wey one (225) 


This equation is independent of the radius and may be 
used to check a second equation as follows: 


JP =HM —IH — PN 
or 
JP = (r+ 34g) sin F — (r+ 3g) sin S —f vers F (226) 


The length of the curved rail KM is found by the formula 
e KM = .017453 (r + 39) (F — S) (227) 
in which the angle or arc (F—S) is expressed in degrees. 


192. Tables for Turnouts. The leads computed by eqs. 
(222) and (223) are called ‘‘ theoretical’ leads. In order 
to avoid the cutting and loss of rails, the American Railway 
Engineering Association has adopted certain combinations 
of switches and frogs. Calculations have been made for 


158 FIELD ENGINEERING 


these combinations and they are given in Tables XXIIA 
and XXIIB. The tangent adjacent to the heel of switch 
is Jengthened in some cases, and in others, that adjacent to 
the toe of the frog. 

The effect of lengthening the tangent adjacent to the 
heel of switch is to lengthen the lead, while that of extending - 
the tangent adjacent to the toe of the frog is to shorten. 
the lead. 





iG. 72. Fie. 73. 


In the next two sections there will be found the effects 
of these two changes on the length of the lead and on the 
radius of the connecting curve. 

193. Given: the theoretical lead, the increase in lead 
necessary for the practical lead and the data for the turn- 
out, to find the length of the tangent adjacent to the 
heel of switch. Fig. 72. 

The other given quantities are F, S, /, g, Li, f and h. 


TURNOUTS AND CROSSINGS 159 * 


Let AG be the theoretical lead; A’G the practical lead; 
AKMF the original turnout; and A’K’’MF that for the 
practical lead. Designate A’K” by Il’. Draw K’K parallel 
to AG and connect K and M. 

The length of switch rail is the same in the two turn- 
outs as is the distance gauge to gauge at heel of switch, con- 
sequently the central angles of the two ares will each be 
equal to (F — S) and the line KM must pass through the 
point K’’ (§ 137) 


A’A = K’'K = the given increase in lead 
KVR KR = Ss K'K"K =4(F = 8) 
and 
K’ KK’ = 180°— (K’K"K + K”K’'K) = 180° — 4(F — S) 
Solvé the triangle K’’"KK’ and 


A’A sin 3(F +8). 


BS hah oi. AC AS) 


(228) 
The perpendicular distance from K’ to the line K’K 
produced is (l’ — l) sin S and if the numerator of eq. (224) 
is reduced by this value we get the radius of the new turn- 
out curve 
—h—fsin F — (l’—1)sinS 
(7 +49) =2 i Ee 


Fad OA) sin (PE 8) CEN 


194. Given: the theoretical lead, the decrease in lead 
necessary for the practical lead and the data of the turn- 
out, to find the length of the tangent adjacent to the 
toe of the frog. Fig. 73. 

The given quantities are the same as in § 1938. 

Designate FP’ by f’. Now PP’ = A’A. 

» Solve the triangle MP’P in a similar way to that in the 
previous section. 
yee od , _ A’A sin 3(F +S) 
LR eel sin 3(F — 8S) 


Also if the numerator of eq. (224) is reduced by (f — f’) 
sin F it becomes g — h — f’ sin F giving the radius of the 
new turnout curve 


(r + 49) = 


(230) 


g—h-—f'snF 


2 sin 3(F + S) sin $(F — 8S) 2st) 


« 160 FIELD ENGINEERING 


195. In case of a double turnout from the same switch 
three frogs are required, and the switch is called a three- | 
throw switch, because its points take three different posi- 
tions. When split switches are used for such a combination 
it is advisable to have the second set of switch rails placed 
slightly in advance of the first set. This form is termed a 
tandem switch. 

Given: the equal frog angles F and F' of a tandem split 
switch, the distance d from one point of switch to the other, 
the radius r of the two curves, and the length l of both switch 
rails, to find the angle of the croteh frog F”’, and the 
crotch frog distance. Fig. 74. 





Fia, 74. 


The rails of the main track frogs and the switch rails will 
be considered straight, while the crotch frog rails will be taken 
as curved. 

Let AKF and BK’F’ be the outer rails of the double turn- 
out, whose radi, CF” and C’F” =r + 39. . 

Produce each curve backward until it becomes parallel 
to the main track rails at the points Gand G’. Draw C’@’ 
to meet CP drawn parallel to AF’. Designate the distances 
from G and G’ to the main track rails by 0, and from the 
figure 

o =h — (r + 3g) versS (232) 


where h = distance gauge to gauge at heel of switch, and 
S = the switch angle. 


TURNOUTS AND CROSSINGS 161 


Solve the right triangle CPC’ for CC’P and CC’ where 
C= GN = AD =d 
PC’ = CG + C’G’ — G'N 
= (r + 29) + (r + 39) — (g — 20) = 2(r+o) 

Then in the isosceles triangle CF’’C’ 
3CC’ — 3CC" 
CEE ort hag 
and the crotch frog F’’ = 2F’’CC’. 


The crotch frog distance DH = F’S + SM. 
Now 


and 


COs. CC = 





(233) 


F”’S = (r + 3g) sin F’C'S 
SM =1—(r+g) sin S 
DH = (r + 3g) sin F’C’S +1 — (r +49) sinS (234) 


The angle F’’C’S is the sum of F’’C’C and CC’P, which angles 
were obtained by solving the triangles CC’F”’ and CPC’. 


~ and 


Where a stub switch is used with the crotch frog sym- 
metrically placed with respect to F and F’, the number and 
angle of the crotch frog are given in Table XXII. 

196. Stub Switch. With the stub switch, the line 
_ from the heel of the switch to the point of frog is considered 
a simple curve. 

Given: a straight track of gauge g, the frog angle F, the frog 
number n, and the throw t, to find the lead of the stub switch, 
the radius Y¥ of the center line of the turnout curve and the 
length of the switch raill. Fig. 75. 

Let C be the center of the turnout; 

F = frog angle; 
g = gauge of track; 

DK = the throw of the switch; 
y = the radius Ca, . 

Two sets of formulas may be found, the one in terms of 
F and the other in terms of n. 


First, formulas containing F. From the figure, 
AB =g = FC vers FCB 


or 
g = (r + 39) vers F 


162 FIELD ENGINEERING 





whence 
Te paar 
(r + 39) Saath (235) 
BF = Ls = AB cot BFA = g cot 4F (236) 
Also 


BF = Ls = FC sin FCB = (r + 3g) sin F (237) 


From the isosceles triangle aCf 


af =)2r sin $F (238) 
Similar to eq. (235) we have 
DK DK 
vers ACD => "ews = (r + 39) 


Also 
AD = (r + 4g) sin ACD. 





Fie, 75. 


But since the inside rail has the same throw as AD, while 
its radius is (r — 3g), we may drop the 3g, hence the length 
of the switch rail is 


AD =r sin ACD (approx.) (239) 

Second, formulas containing n. 

From the triangle BFA, 
BF = Ls = AB cot BFA = g cot 3F 

But from eq. (220) cot 4F = 2n 
ae BF = Ls = 2gn (240) 
From the triangle BFC 
BF? = FC? — BC? 


TURNOUTS AND CROSSINGS 163 


or 
BF? = Ls? = (r + 39)? — (r — 39)? = 4g?n? 
whence 
2gr = 4g?n? 
or 
ri=i2on7 = Dyn? - (241) 


Draw the line CM bisecting af. Then the triangle aCM 
and ABF are similar, and 


aM aC VAD SAF 





or 
tafe PAL AR 
. SED 
ee af = AP 
but. 
AF = VBA? + BF? = Vg? + 4g?n? = gV1 + 4n? 
consequently 
if ee (242) 
V1 + 4n? 
2 
From eq. (38) the throw DK = t = 
l= AD = V 2rt (243) 


Table X XII has been calculated by the preceding formulas. 
This table applies only to stub switch turnouts. 


197. The rails between the switch point and the frog 
point may be located in three different ways: 

First, decide upon the position for the frog point and lay 
off the proper lead, thus locating the switch point. Place 
_ the switch rails and the frog in position and locate the con- 
necting rails by eye. 

Second, offsets may be taken from a gauge side of the main 
track rail to the gauge side of the turnout rail. Fig. 76 applies 
to a split switch. The curve KM is produced backward 
through the angle S where it becomes parallel to the main 
track rail AD. Divide KM into four parts and through 
each point of ‘division draw lines perpendicular to AD. 

From eq. (2382) 


164 FIELD ENGINEERING 


VW =o0 =DK — (r+ 3g) vers S =h — (r + 3g) versS 
Angle KCM = (F —S) 


WCS=S+i(F —S) WCT =S+3(F —S) 
wcUuU =S+i(F —S) WCM =S+(F —-S) = 

Then AD = | approx., and 
VA = VD—AD= (r+ 3g) snS8 —l=q 


eee) 








es 


Center Line of Main Track 


ee 
yom 
“Sy 








TGs (om 


Note that V will fall between A and D with long switch 
rails. 


Now 


= (r + 3g) sin WCS — q 
and 


S'S = (r + 3g) vers WCS + 0 
Similar expressions may be given for AT” and T’T, AU’ 
and U’U and so on. 


Table XXIIB gives the coordinates of the BEEN points 
of the different lead curves. 


TURNOUTS AND CROSSINGS 165 


With a stub switch the offsets may. be found by the methods 
of §§ 119 and 120. 

When a tangent is introduced beyond the heel of the 
switch the value of o from eq. (232) will be 


o=h+(U —1) sin S — (r+ 4g) vers S (232A) 


Third, having placed the switch rails and the frog, the 
lead curve connecting the two may be located by taking 
offsets from the long chord of the curve. The length of the 
long chord of the outer rail of a split switch is deduced from 
Fig. 76, and is KM = 2(r + 3g) sin 3(F — 8S). The values 
of the offsets may be found by the methods of §§121 to 
. 125. As applied to the stub switch, the middle-ordinate 
of the long chord AF, Fig. 75, is +g, and the ordinates at the 
quarter points are 75g. 

198. The track beyond the heel of the frog will assume one 
of the following forms: It may be straight, or there may be 
a combination of curves and tangents; it may be part of 
a cross-over between parallel tracks; or it may be extended 
to a parallel side track. 

No special formulas need be derived where there is no 
definite connection with a cross-over or parallel side track. 
The method of procedure is illustrated by Fig. 77. <A 
tangent at the frog point is located by sighting along the 
line F’G with the angle F laid off on the proper side of zero. 
Aftor the location of this tangent, the work proceeds as 
in any straight line or curve work. 

199. Cross-overs. A cross-over is a track connecting 
two parallel tracks, either straight or curved. ‘There are 
two usual forms, first, where the heels of the frog are con- 
nected by a tangent, and second, where they are connected 
by a reversed curve. 

Given: the perpendicular distance p between two parallel 
straight tracks, the gauge g, the equal frog angles F and EF", to 
find the length of tangent d, between the frog points. Fig. 78. 

Let AF be one turnout, and A’F’ the other, connected by 
the straight track GF’: Producing the line F’G to inter- 
sect the rail NF at H, we have two right-angled triangles, 
GFH and F’NH, having the common angle at H = F. Let 
p = the perpendicular distance between center lines of the 


166 FIELD ENGINEERING 


main tracks, and g = the gauge. Then GF = g, and F’N = 
(p — 9). 


FEC. Fi (egg ee pote 
sin FH 
or 
PG =d =" —F —g cot F (244) 
Also 





sin 


FN = NH — FH = (p —g) cot F — - (245) 





‘ 
‘ 
‘ 
4 
i 
' 
‘ 
| 
‘ 
‘ 
‘ 
‘ 
1 
‘ 
' 
' 
' 
' 





Fig. 79. 


It is to be noted that d in eq. (244) is the distance from the 
theoretical frog point at F’ to the point G opposite the theo- 
retical point F. Also that the distances d and FN are 
independent of the shape of the rails between the frog points 
and switch points. 

The distance between the actual frog points, measured 
along the main tracks will be FN — 2nt, where? = the width 
of the frog point. 

200. Given: the perpendicular distance p between the parallel 
straight tracks, the frog angles F and F’, the length of the frog 
rails from point to heel k and k’, the radii r; and 12, of the two 


TURNOUTS AND CROSSINGS 167 


arcs of a reversed curve connecting the heels of the frogs, to find 
the central angles of the two arcs. Fig. 79. 

Let CP = 1, C’P =n, FM =k, F’N =k’. 

Produce the outer rails of-the two ares of the reversed 
curves backward until they become parallel to the main 
tracks. Also drdw MM” and NN” perpendicular to CA 
and C’A’, and FM’ and F’N’ perpendicular to MM” and 
NN” yespectively. : 


Now 
AG =o, = AB+ FM’ —GM" 
or 
o =¢g+ksin F — (r + 3g) vers F (246) 
Similarly, 
0, =g +k’ sin F’ — (r + 4g) vers F’ (247) 
* Consequently from eq. (209) 
ACR = ps 
vers AC Ata (248) 


Then 
MCP = ACP — F and NC'P’ = A’C'P — F’ 
When the two frogs are of the same number F = F’, 
1 = 12 = 7, and 0; = 02 = o and eq. (248) becomes 


1 = 
vers AGP = 222 (249) 


r 
MCP = NC'P = ACP —F 


201. Turnout to a Parallel Side Track. Given: a frog 
angle F, and the perpendicular distance p between the center 
lines of the main and side tracks, to find the radius r’ of the 
curve connecting the turnout with the side track. 
Fig. 80. 

Let the reversing point be taken at the heel of the frog 
and let Q be the center of the required curve. Draw QM 
perpendicular to the main track. Denote the length of the 
frog from point to heel by k. The angle at Q will equal F. 


and 


Now 
MS = MN —SN 
or 
(r’ — 4g) vers F = p—g—ksinF 
o. igang ales anf (250) 


vers F 


168 FIELD ENGINEERING 


Also 
FN = FT+TN =kcosF + (r’ — 39) sin F (251) 
When 7’ is fixed by a limit, we obtain k by resolving eq. (250) 


kp sig erg) ivers F 
rs sin F 





k 


202. Ladder Tracks. A ladder track is one that leads 
from the main yard track to a series of parallel tracks. 
The parallel tracks of a yard are called body tracks. 

There are two usual forms of ladder tracks, first, where 
the ladder is straight beyond the heel of frog, and second, 
where a short curve is inserted beyond the heel of the frog. 

Given; a ladder track making an angle F with the main 





track and straight beyond the heel of the frog, to find the dis- 

tances between the theoretical frog points. Fig. 81. 
Let the perpendicular distances between the tracks be 

denoted by p, p’, ete. 

Now 


F’'G = p, FG’ = p’ and so on. 
Then 


Py eae Ate Le p’ 
Ae kc sin F ang Ela sin Ff 








It is to be noted in this case that the character of the turn- 
but curve does not affect the solution. The turnout may be 
of any desired form. 

The distance between frog points on a ladder track should 
be large enough to give sufficient space for the lead, the dis- 
tance from point to heel of frog, space for the angle splice 
at the end of the frog, and a certain extra allowance for 


TURNOUTS AND CROSSINGS 169 


clearance between the splice and the switch including that for 
the necessary bending of the stock rail. 

203. The body tracks of a yard may be slightly lengthened 
by inserting short curves beyond the heels of the frogs of 
the ladder track. 

Given: the ladder and body tracks as shown in Fig. 82, the 
radius of the curve beyond the heels of the frogs, 











TG. 


the equal frogs F, F’ and F’’, to find the angle MCN and the 
distances between the frog points. 

Let r denote the radius of the given curve. 

Produce the curves backward from the heels of the frogs. 
The offsets MM’, NN’, etc., are equal to o and may be found 
from eq. (246). 

If the are M’SN be moved parallel to itself until M’ 
and M coincide, then N and N’ will nearly coincide and a 


170 FIELD ENGINEERING 


reversed curve will be formed in which p and the radii are 
given. 
Now 

CN’ =r+34g +a and C’N’ =r — ig 


Consequently from eq. (209) 


vers NCM = (252) 


peeobt epg alee 
CN+C'N* 2r-Fa 


From § 202 
aise. De tae! 
sin MCN 


NP = 


Since equal frogs are used, NF’ = PF’. Adding the dis- 
tance F'’P to both sides of this equation there results 


IN Bigs 4 Pees 
ome RE wae By (ohh fe Coe 

204. Turnouts from Curved Tracks. When a turn- 
out leaves a main track, the parts of the frog on the turnout 
may be considered straight, but inasmuch as the parts forming 
sections of the main line are arcs of a curve, it seems an 
unnecessary refinement to differentiate between the two 
sides of the frog. 

Consequently equations will be developed considering 
both switch and frog rails curved as they would be con- 
sidered with a stub switch. 

Given: a curved main track 
and a frog angle F, to locate 
a turnout on the inside of the 
curve. Fig. 83. 

Let R = Oa = radius of main 
track, r = Ca = radius of turn- 
out, and F = CFO = the frog 
angle. 

In the diagram draw the chord 
AF and produce it to intersect 
the outer rail at G; and draw 
FO and GO. Since the chords AF and AG coincide, and 
the radii AC and AO coincide, the chords subtend equal 
angles at C and O respectively, and GO is parallel to FC. 
(See § 137.) Hence, FOG = CFO = F.° Let 0 = the angle 
FOA. 





TURNOUTS AND CROSSINGS TEL 


In the triangle FOA, 6 = GFO — FAO = GFO — FGO; 
and in the triangle GFO, GO-+ FO : GO — FO :: tan 
4(GFO + FGO) : tan 4(GFO — FGO), or 2R : g :: cot 
2F : tan $0. 
vit 





tan $0 = @ cot 3F = R (254) 

In the triangle CFO, 

neAt ai sind 

In the triangle BOF, 

BF = 2(R — $9) sin 30 (256) 
In the triangle aCf, 
af = 2r sin ($(F + 6) (257) 
Example.—Let R = 1432.69 and F = 6° 43’ 59”. 
Eq. (254) 39 2.354 log 0.371806 
if 3° 21’ 59.5 Jog cot 1.230440 
log 1.602246 
Rk (Table I) ee De LoGlLS | 
19 — -1°35/ 59.8 log tan 8.446095 
Ka. (255) 6 a. 44%59'°6 sin © 89746086 
r+ 6 OD OS 40 bees sO oO te 
9.510008 
R—ig 1480.336 3.155438 
r+ig 462.856 2.665446 
r 460.502 

(256) 2 log 0:301030 
(R — 3g) 14380.336 EPS E1DO4OS 
30 1° sv 59.8 logsin 8.445924 
Et BF 79.872 log 1.902392 
(257) 2r 921.004 ‘2.964262 
4(F + 6) 4° 57’ 59.3 = logsin 8.937381 
af 79.734 log 1.901643 


The values of BF and af are found to be so nearly identical 
in this case with those determined in case of a turnout from 
a straight line, that the values given in Table XXII may 


172 FIELD ENGINEERING 


be used at once for ordinary values of R; and the degree of 
curve of the turnout in this problem is approximately the 
sum of the degree of curve of the main track and the degree 





Fig, 84. 


of curve given in Table XXII opposite F. Thus, in the 
example 4° + 8° 26’ = 12° 26’... r = 461. 7 nearly. 
Expressed in an ste there results 


=D+d (258) 


where d’ = the degree of curve of the new turnout, D = the 


(approx.) 





Fie. 85. 


degree of the main track and 
d =the degree of curve of a 
stub switch turnout from a 
straight track and for the same 
frog number. 

205. Other cases may arise 
as shown in Figs. 84 and 85. 

In Fig. 84, BF is the curved 
main track and AF the turn- 
out is on the outside of the 
curve. - 


By a similar reasoning to that given in the previous 
section, and with the same notation it may be proved that 


tan 30 = 


aR 


tgp 45,9 
cot $F R 


which is identical with eq. 258) 


TURNOUTS AND CROSSINGS 173 


In CFO, 
(r + 39) = (R + 39) Se 
In BOF,. 
BF = 2(R + 3g) sin 46 
In af, 


af = 2r sin 3(F — @) 
(approx.) d’=d—D (259) 


When d = D, the turnout becomes a straight line. 

In Fig. 85, the turnout AF is on the outside of the main 
track but its center C falls on the same side as O. The 
equation for this case is 

sin 0 


PSO aD eg = 


BF = 2(R + 39) sin 40 
af = 2r sin 4(6 — F) 
(approx.) d’ = D—d (260) 


206. Split Switch Turnouts from Curved Tracks. 
It may be proved in a somewhat similar manner to that 
outlined in the previous sections that for a certain frog 
number the lead for a split switch turnout going from a 
curved track is practically the same as the lead from a 
straight track of the same number. The degrees of curve 
“are expressed approximately by eqs. (258), (259) and (260). 

Tables XXIIA and XXIIB give the standard split switch 
data. 

207. To Connect a Curved Main Track with a 
Parallel Siding. 

Given: the perpendicular distance p between the center 
lines of a curved main track and a parallel side track, and the 
frog angle F of a turnout; to find the radius Y of the con- 
necting curve, and the length FN, or fm, of the curve. Fig. 86. 

Let FN be the rail of the main track, and GM the rail of 
the siding, adjacent to each other; let O be the center of the 
main track, and Q the center of the connecting curve. Then 
the connecting curve will terminate at m, on the line OQ 
produced. 

In the diagram draw MF, and produce it to intersect the 
rail MG at G, and join GO, FO, and FQ. 

Let R = radius of center line of the main track; r = 


174 FIELD ENGINEERING 


radius of center line of the connecting curve; and @ = the 
angle FOM. 
Case a.—The siding outside the main track. Fig. 86. 
By similarity of the triangles GOM and FQM, GO is 
parallel to FQ, and the angle GOF = F; and by a process 
similar to that of § 204 we have from the triangle FMO 





tan 46 = Has cot 4F (261) 
From the triangle FQO 
sin 6 . 
r—3g = (R+ 349g) ———— 262 
29 29 sin (F of. @) ( 4) 
FN = 2(R + 3g) sin 46 (263) 
fm = 2r-sin 4(F + @) (264) 





Case b.—The siding inside the main track with the center 
of the connecting curve outside the main track. Fig. 87. 
By a process entirely similar to § 204, we have 





tan 36 = en cot 4F (265) 
sin @ 

r— 3g = (R — 3g) Se (266) 

FN = 2(R — 3g) sin 30 (267) 

fm = 2r sin 3(F — 6) (268) 


When 6=F in the last equations, sin (F — 6) =0, 
and r — 3g is infinite, and the curve FM becomes a straight 
line. 

Case c.—The siding inside the main track with the center 
of the connecting curve inside the main track. In this case 
6 will be greater than F, 


TURNOUTS AND CROSSINGS 175 


The resulting equations will be 
i 1 si 6 > 
| r+40-(R- Way (269) 
: fm = 2r sin 3(6 — F) (270) 


Eqs. (265) and (267) remained unchanged. 

208. Given: the perpendicular distance p between two 
parallel curved tracks, to connect them by a cross-over in 
_ the form of a reversed curve. Fig. 88. 

Let O be the center of the main curve, and C and C’ the 
centers of the reversed curves. The frog rails will be con- 


















sidered curved throughout. The ares of the reversed curve 
are produced backward as shown. 

Two cases may arise: first, where the two radii of the 
‘reversed curve are assumed; and second, where the frog 
numbers or frog angles are assumed. 

Case I. Assuming r, and r2, the radii of the ares beyond 
the frog. 

_ Find the values of the offsets 0; and 0; at A and A’ from 
the general formula 

o =g — (r + ig) vers F (271) 


This formula is derived similarly to eq. (246), but in this 
ease, the frog is considered curved from theoretical point 
to heel. : 

Then in the triangle COC’ we know all three sides; for 
CO=Rinta; CC’ =n-+n; and CO=Rip— 
72 — Gz. 


176 FIELD ENGINEERING 


Solve for the angles COC’ and C’CO. 

The angle C’CO = C, determines the length of the are 
described with the radius 7; the angle (0 + C) =CC’A’ | 
determines the length of the second arc, and P is the point 
of reversed curve. 

Case II. Assume n and n', or F and F’. Take from 
Tables XXII, XXIIA, or XXIIB, the degree of the turn- 
out curve from a straight track for the particular frog number. 

Find from eqs. (258), (259), or (260), the degrees of the 
turnouts from the curved tracks. The radii corresponding 
to these degrees are found from Table I and the value of 
0, and 02 from eq. (271). 








Fig. 89. 


The solution from this point is the same as in Case I. 

The particular advantage of this case lies in the fact that 
regular frogs are used, while in the other special frogs may be 
necessary on account of the arbitrary selection of the 
radii. | 

B. Crossings. 

209. When two tracks cross, four frogs are necessary. 
Both tracks may be straight, one may be straight and the 
other curved, or both may be curved. With both tracks 
straight, two frog angles are supplements of the other two. 
In this case, right triangles are solved and no special formu- 
las need be derived. 

Given: a straight track of gauge g§ crossing a curved track 
utth radius BR and gauge g’,and the angle between the 
center lines, fo find the frog angles and thed istances 
between the frog points. Fig. 89. 

From O, the center of the main track, draw the lines OF, 
OF’, OF”, and OF’; also OH perpendicular to VH. 


TURNOUTS AND CROSSINGS 177 


Let g and g’ be the gauges of the straight and curved tracks; 
OV = R, and angle HVO = 90° — A. 
Then 
OH = OV sin (90° ~ A) = Reos A 
OH —3 
_ OF 29 
pele 20: 
_ OH +39 
fi — 39’ 
fs OH — 3g 
Rh — 39’ 
IR = FOK = FUCK 
= (R + 3g’) sin F’ — (R — 3g’) sin F” (276) 
A similar equation may be found for FF’. 
RR = (+ 39’) are (fF — F’) (277) 


”) 


cos ff = (272) 





cos F’ (273) 


cos F’’ (274) 


cos F’”’ Calas), 


in which “arc ”’ is regarded as a function of the angle, to 
radius unity, and to be found in Table XX. 


So also, 
PE AL 19g.) are (i! P"') (278) 


It is to be noted that these lengths are between the theo- 
retical frog points. In constructing the crossing allowance 
should bé made for the flangeways. 

210. Given: the radi R and R’ of two curved tracks of 
gauges & and g’ which cross at V at an angle A, to find the frog 
angles and the length of the curved rails. Fig. 90. 

Let OV = R, O'V = R’, and OVO’ = A. 

Solve the triangle OVO’ for OO’ having given two sides 
and an included angle. The triangle OF,O’ may be solved 
for fF, and the angles F,OO’ and F,0’O, having given 
OF, = R — 3g, O'F; = R’.+ 49’, and OO’. 

Similarly with the triangles OF 0’, OF;O’ and OF 40’. 

Then the angle 

FOF, — F,OO’ = F,0O' 
and 
FF, = (R = 39) arc FOF». 


The other three lengths may be calculated in a similar 
way. 


178 FIELD ENGINEERING 


211. Given: two straight tracks crossing at an angle A, 
two equal frogs F and F’, and'the radius R of the simple curve 
connecting the heels of the frogs, to find the distance from 
the center of the crossing to points opposite the 
theoretical frog points. Fig. 91. 

Let O be the center of the connecting curve. Designate 
the intersection of the inside straight rails by X. Produce 
the connecting curves backward as shown. 





Fia. 90. Fig. 91. 


The offsets, 0, at B and B’ may be found from eq. (246). 
The central angle MON = A — F — F’ =A — 2F, 


Now 
OB = OB’ =R-+0—ig 
and 
XB = XB’ = (R +0 — 3g) tan 3A 
From the figure, 
BF = B'F’ = (R + 3g) sin F —k cos F. 
Then 
XF = XF' = XB — BF. 


For the center line distance from V to points opposite 
F and F’, add 3g tan $A to each. 

nt should be subtracted from each to locate the actual 
frog points. 


The length of the are MN = (R + $9) are (A — 2F),. 


TURNOUTS AND CROSSINGS 


212. Given: two straight tracks crossing at an angle A, 


179 


and the location of one frog, to connect them by a turnout in the 
form of a reversed curve. 


Fig. 92. 





. 


len 





1 
=! 


the frogs from points to heels. 
CM = Ry +- a0. 


Let F and F’ be the frog angles, k and k’ the lengths of 
The distance F’V is given. 


Also C’N = R; + 3g, and 


180 FIELD ENGINEERING 


Produce the ares of the reversed curve backward until 
they become parallel to the main tracks. Draw the other 
construction lines as shown. Then 

F'G = F’V sin A + 3g 

NF'N’ = 180° — (90° — A) — F’ = 90° — (F’ — A) 

Now 
F'N’ = F'N cos NF'N’ = k’ sin (F’ — A) 


and 
NN’ = k’ cos (F’ — A) 
NH = F'G — F'N' = F'V sin A + 3g — k’ sin (F’— A) 
Also 
LE =NH+ NC’ vers NC’L 
= NH + (Rez + 3g) vers (F’ — A) 
= F’V sin A + 3g — k’ sin (F’ — A) + (R2. + 39) 
vers (F’ — A) 
GE = HE — NN’ 
or 


GE = (R2. + 4g) sin (F’ — A) — k’ cos (F’ — A) (279) 


Now LC’ — (4g + 01) will be the perpendicular distance 
p between parallel tracks; 0; is found from eq. (246). 


from eq. (209) 


pe sumed 7 
vers ACP = Rak, 
Then 
MCP = ACP — FandPC'N = PC'L — (F’ — A) 


EA = (Ri + R2) sin ACP and the distance from G to 
the point opposite F is 


EA — EG — [((Ri + 39) sin F—k cos F}. 


Since the central angles of the arcs have been found, their 
lengths may be obtained. The position of the frog F is 
fixed by its distance along the center line from the point G. 

213. At the crossing of two tracks, traffic may be switched 
from one track to the other by a combination crossing called 
a Slip switch. The switches and lead rails are entirely 
included between the main frogs. A great saving of room 
is gained over other forms of connections, but on account 
of the small distance between the extreme frog points, the 
angle between the tracks must be slight, 


TURNOUTS AND CROSSINGS 181 


Given: the angle at which two straight tracks cross and the data 
for the switches and frogs, to find the distances along the curved 
rail between frog points for a slip switch crossing, and the 
radius of the curve connecting the heels of the switches.. Fig. 93. 

Let AK = BJ = the length of the switch rail, 1, and DK 
and LJ = the distance gauge to gauge at heel of switch h. 


' 
1 
' 
1 
1 
| 
4 
' 
| 
oe ~ 
! 
! 
1 
4 
! 
1 
| 
' 
| 
i} 
| 





The distances F;A and F;B are governed by the clearance 
necessary from frog point to switch point as mentioned in 
§ 202. 

The distances between successive theoretical frog points 
are all the same and are expressed by 


Fibs = Moh. = beh, = Wibys (280) 


ba A 
sin Ff 
At the point Ff, and F, the theoretical and actual points 
coincide. 
The direct distance between theoretical frog points Fi; 
. and F’3 is 


v Pip ee (281) 


182 FIELD ENGINEERING 


Similar to the solution given in § 191, two expressions 
for JT’ may be found which are 
JT =g —h—F,A sin F —lIsin (F — 8S) 
and 
JT = (R + 39) cos S — (R + 39) cos (F — S) 


C 





Fig. 94. 


Equating these two values and solving for R we obtain 
g—h—F,A sin F —l sin (F —S) 
cos S — cos (F — S) 


The length of the outer curve is 
KJ = (R + 3g) are (F — 28) 


214. Given: a straight track, the frog numbers and turnout 
data, to locate a symmetrical Y track. Fig. 94. 

Let CM and C’N = R + 3g = the radii of the outer rails 
of the curves connecting the heels of the frogs. 

Draw CA and C’B perpendicular to the straight track; 
MM’ and NN’ perpendicular to CA and C’B. Draw the 
other construction lines as shown. 


Then 
CA =CM’'+M’'A =(R+ 39) cosaF+3¢g+ksin F 
GO =k’ sin 3F” 


ieee (282) 


TURNOUTS AND CROSSINGS 183 


and CO = (R+4g) +k’ sin 4F” 
MCO = 90° — F — }F” 
The solution depends on solving several triangles. 


First, ACO for AO and angle CAO, knowing two sides 
and the included angle; second, COE for OF. knowing CO 





and ECO = 3F”; third, AVO for VA, knowing AO and 
VO =CA — OE. 


Finally 
VP =VA—PA =VA — (R+ 3g) sn F+kcosF 
and 
VF" = VO+0F” =VO+K’' see 4F” 


Since all the frog points are located with respect to V, 
the problem is solved. 

215. Given the frog angles fora Y between two diverging 
straight tracks, one curve being tangent to the heel of a frog 
in each of the straight tracks, and the second being tangent to 


184 FIELD ENGINEERING 


the center line of the other, to find the position of the frogs. 
Fig. 95. 

Let AB and NE be the, center lines of the given straight 
tracks intersecting at V with an angle A, MN and M’'N’ 
be the connecting curves, 

Produce the line AM, a tangent to the curve, to V’ 
where it meets HN produced. 

If a line F'A were drawn it would bisect the Rete F, Then 
PA = 39 cot $F = gn. 

AM =gn+kand AV’ =gn+k+R tan 3(A — F) 

In the triangle AV’V 

AV : AV’ :: sn F : sin (A — F) 
or 
AV’ sin F 
sin (A — F ) 
VP =VA—PA=VA-—gn 


AV = 





This equation locates the point P which is opposite the 
frog point F. The positions of F’ and F” are located in the 
manner outlined in § 211. 


CHAPTER IX 
THE SPIRAL CURVE 


216. In the early years of railroad building it was con- 
sidered quite sufficient that the curves should be circles 
and the straight lines truly tangent to the curves. But as 
train speeds were increased it became necessary to intro- 
duce a curve of variable radius between the tangent and circle 
for two reasons: First, to avoid the lateral shock pro- 
duced at the tangent point either on entering or leaving 
the circle, and second, to permit an elevation of the outer 
rail which might be everywhere proportional to the degree 
of curve. See Chapter XVII for derivation of formula 
for elevation of the outer rail. 

A car running at high speed on a straight track is sud- 
denly forced aside and takes up a circular motion on striking 
a curve, and when upon the curve the car body continues 
to revolve horizontally, like a huge bar suspended at its 
center of gravity, until its revolution is suddenly arrested 
by another piece of straight track. In either case a shock 
is produced deleterious both to track and rolling stock. 
By introducing a short curve of variable radius the transi- 
tion is made easy and almost insensible. 

The spiral curve here described fully meets the above 
requirements. It not only blends with the tangent at one 
end and with the circle at the other, but it also affords 
sufficient distance between the two for the gradual elevation 
of the outer rail. In shape it conforms very nearly to the 
cubic parabola with the advantage of requiring less track 
near the tangent point. The first radius is amply long 
without being infinite. The spiral curve is constructed 
upon a series of equal chords, and the angle subtended by 
the first chord is made the common difference for the angles 
subtended by the succeeding chords. It is a multi-compound 

185 


186 FIELD ENGINEERING 


curve in which the degree of curve progresses in arithmetical 
ratio from chord to chord. 

217. Let the chord length be 100 feet and the central 
angles be 10’, 20’, 30’, etc., subtended by these chords. 
Then these numbers represent the degrees of curve on the 
several chords, and their sum the total spiral angle up to 
any specified point from the tangent point S. The in- 
clination to the tangent of any one of these chords is evi- 





Fie. 96. Fie. 97. 


dently equal to half the angle it subtends added to the total 
preceding central angle. Thus we have, if n be the serial 
number of chord, and s the spiral angle: 


n i 2 3 4 5 ete. 
sub. 10’ 20’ 30’ 40’ 50’ ete. 
Ss 10’ aU. 8 007 Ay oS eee 


inclination 5’ AW £5 I a Dk ete, 
_ Thus we obtain the first four columns of Table XXXIV. 


Taking the tangent as a meridian, the inclination of a 
chord becomes its bearing, from which its latitude, or 100 


THE SPIRAL CURVE 187 


eos incl., and departure, or 100 sin incl., may be quickly 
computed. Taking the tangent point S as an origin, and 
calling x and y the coordinates of the several chord-points, 
the continuous summation of latitudes gives values of y 
and of departures gives values of x. These are all tabulated 
in Table XXXIV. 

218. Spiral Deflections. Fig. 96. With the transit 
at S the deflection 7 from the tangent to any other chord 
point is found by the formula 


er Si 
tant = - 283 
ji (283) 


With the transit at any chord point, n, to find the deflec- 
tion from the local tangent through nto any other chord 
point: observe, first, that for the adjacent point, whether 
forward or backward, the deflection is one half the angle 
subtended by the adjacent chord; thus Fig. 97, with transit 
at point (3) the deflection to (4) is 20’ and to point (2). it 
is 15’. For more distant forward points, draw long chords 
to them from n and find the angle a that such long chord 
makes with the main tangent, and deduct the angle between 
the main and local tangents; the difference is the value of 
irequired. The angle a is found from 

xc — x’ 


tan a =- ; 284 
yey a 





in which z, y are the coordinates of the point observed 
and x’, y’ of the position of transit. And 


t+=a-—s (285) 


Thus, with transit at (3) we have for point (5) 


7.998 — 2.036 5.962 
= “ ~ 02982 
tan 4 = 799 896 — 299.980 ~ 199.907 ~ 079825 





and 

a = 1° 42! 30” 
Since for (38) se 1° 

4.=,00° 42’ 307 


The angle subtended by a long chord is equal to the differ- 
ence of the values of s for the extremities of the chord; it 
is also equal’'to the sum of the deflections made at either 
end of the long chord to find the other. Hence, knowing 


188 FIELD ENGINEERING 


the central angle and forward deflection along any long 
chord, we find by subtraction the backward deflection 
along the same chord. Thus, with transit at (3) to find 
(1), since s’ — s = 50’ and the forward deflection from (1) 
is 22’ 30’, the difference, or 27’ 30” = 7% = the deflection 
required. 

In this manner the entire Deflection Table XXXYV has 
been calculated, and since the Spiral is invariable in form 
all its angles are constant and the deflections given are 
available in every case that can occur in practice. 

219. To adapt the Spiral to practical require- 
ments it is only necessary to draw the figure to a different 
scale. All its dimensions remain proportional to each other 
including the radii of the ares composing the Spiral. The 
chord length assumed at 100 feet is now to be given smaller 
values ranging from 10 feet to 50 feet, and all other parts 
are calculated accordingly, varying directly as the chord 
length, while the degree of curve varies inversely, or nearly 
so. Table XXXVI furnishes exact values of the degree 
of curve, Ds, length of Spiral, nc, and values of the coor- 
dinates at each point for a variety of chord-lengths. Among 
these Spirals may always be found one or more that will 
be well adapted to the case in hand. 

Only so many chords of a spiral will be used as will lead 
up to the required degree of curve which is to follow. The 
number 7 should be such that the degree of curve on (n + 1) 
will equal, as nearly as may be, the required degree of curve. 
If the length of spiral, nc, is not satisfactory for any reason, 
another selection may be made from the Table under another 
chord length. The diagram shown in Table XXXIII 
gives the recommendations of the Am. Ry. Eng. Assoe. 
for the proper length of spirals. The basis for this diagram 
is the increase per second of the elevation of the outer rail. 

220. To find the spiral long chord. SJL, and the 
Spiral tangents, SH, LH, Fig. 98. 

Let L be the terminal point of the spiral, x, y, the coor- 
dinates of L, i the deflection HSL and s the central angle 
YEL., : 

Then from the figure, 


SE ee He ess (286) 
COS v sin 2? 


THE SPIRAL CURVE 189 


To find the spiral ‘tangents; in the triangle LEY, 


Ni and 4 Sh = y —xecots (287) 
sin s 


To find any other spiral long chord, as KL, and the spiral 
tangents, KH’, LE’; Let s’ be the central angle for K, 
and 7’ the deflection at K for KL. Then from the figure 
TOY er 2 Pia Se 


oe cos LKN cos (s’ — 7’) 





(288) 





Fia. 98. Fig. 99. 


Take values of y and y’ from Table XXXVI; values of s’ 
and 2’ from Table XXXY. 

In the triangle KLE’, since KE’E = (s — 8’), 
KL sin wv’ KL sin i . 
= — KE = 289 
a sin (s — s’)’ sin (s — s’) ee) 

Values of Spiral long chords and tangents are tabulated 
in Table XX XVIII. 

221. With a spiral terminating at L, if the circular arc 
upon the following chord length were produced backward 
from L through the angle s it would terminate at A’ in 


190 FIELD ENGINEERING 


a line parallel to the main tangent, the perpendicular distance 
p, between the two being LM — JA’ or 
p =x — FR’ verss (290) 
in which R’ is the radius of the spiral are (n + 1). 
Also 
q=SM —LI =y-—R’'sins (291) 


We thus have q and p as the coordinates of the point A’ 
which are often convenient either in the field or the office. 
Values of p are tabulated in Table XXXVIII. But if the 





Rie. 100°. 


radius of the curve beyond L differs from that of the spiral 
are (n + 1), these values of p will not apply to it and must be 
computed. (Compare p in Tables XX XVIII and XXXIX.) 

222. Combination of Curve and Spiral. Fig. 100. 
Given: A circular curve with equal spirals joining two tangents, 
to find the tangent distance, T, = VS. 

Let SL be one spiral, and LH one half of the curve with 
its center at O. Join OL and OV, and through L draw GJ 
parallel to the tangent and GN, LM, perpendicular to it. 
Then 

IOL = s, IOV = 3A, OL = R’, SM =y, LM =z 
SV =SM+NV+MN 


THE SPIRAL CURVE 19] 


sin LOG ,sin (3A — s) 
PGT Ge CeinG AN ste ecarhALaS 

Hence 

sin (3A — s) 


Ts =y+-2ztan 4A +4 R’ ee 


(292) 
Example.—Total angle 42°, Degree of curve 7° 20’, Spiral, 
10 chords of 24 feet each. What is the Tangent distance? 
Ans: Ts = 416.24. 
For a small change in FR’ but using the same spiral, what 
will be the change in 75? If in eq. (292) we give R’ two 
values in succession and subtract, we shall have 


,sin (4A — s) 
i cos 3A ar 


diff. Ts = diff. 
When selecting a curve with spirals to suit certain tangents 
use the following Approx. 


T; = R’ tan 4A + inc (294) 


or, to the tangent distance for a simple curve add one half 
the length of an appropriate spiral; the sum will be the 
approximate distance required. When a satisfactory selec- 
tion has been made compute the exact 7's by eq. (292). 

223. Given: A circular curve with equal spirals, joining 
two tangents, to find the external distance, VH = &;, 
Fig. 100. 


In the figure 
VH = VG+GO0 —OH 





Wend p COe Spe 
ss ~ eos 4A taf cos 4A 
_ 2+’ (cos s — cos 2A) 
ag cos 3A (295) 


For a small change in R’, but using the same spiral, the 
corresponding change in EH will be 


cos s — cos 3A 


diff. Hs = diff. R ee 


(296) 





Ezample.—Total angle 42°, Degree of curve 7° 20’, Spiral 
10 chords of 24:feet each, What is the External distance? 
Eq. (295). Ans. Es; = 59.298. 


192 FIELD ENGINEERING 


Example.—Changing the above degree of curve to 7° 30’ 
will make what change in the value of Es? (diff. R’ = 17.351). 
Kq. (296). Ans. diff Es = 0.997. 

Hence new value of Hs = 58.301. 

For a small change in c, the chord length (n and therefore 

s being constant), and with the same radius as before, we 


have from eq. (295) 
aur: 7 


cos 3A 

The change in c must not be so great as to make the last 
degree of curve of the spiral incompatible with the main 
curve. 

Example.—Changing the chord length to 26 feet in the 
above example will make what change in the value of Es? 
CE Sr: aaa 18) Ans, diff. he = TAGZ 

New value of H; = 60.495. 

224. With EH; unchanged and the angles A and s the same, 
it is required to find the change in R’ required by a 
change in x in order to close on the same pin Seer 

Solving eq. (295) for R’ we have 

se pes C08: GA — 2% 
~ cos s — cos 3A 





diff. Hs = (297) 





in which, by giving to v any two values and subtracting one 
result from the other we derive 

— diff. x 
cos s — cos 3A 
The minus sign shows that R’ decreases as x increases and. 
vice versa. 

The new value of z is taken from the Table for the same 
value of n as before, and such that the new spiral and result- 
ing curve shall be consistent with each other. 

Example.—Let D’ = 7° 20’, A = 42°, s = 9°10’, n = 10, 
c = 24. 

Select » = 10, c = 26, Then ‘ diff. «= — 1.118. and 
diff. R’ = 20.84, Ans. 

R’, 781.84 + 20.84 = 802.68, say 7° 09’. 

Note that with new value of x goes a new value of y. 

225. Given: A simple curve, to replace it by another with 
spirals on the samé ground and of same length, as nearly 
as may be, Fig. 101. 





diff. R’ = (298) 


THE SPIRAL CURVE 193 


Let AH be one half the given curve with center at O, 
H’L one half the new curve with center at O’, and SL its 
spiral, LM az, SM =y, HH’ =h, .the radial offset, 
CA’ = p, the clearance of the curve produced. Then 


ON = AO —-NF-—-FA=R-—R' coss—2 





Also 
ON = O00’ cos }A = (R +h — R’) cos 4A 
Fig. 101. 
whence 
R’ (cos s — cos 34) = R vers 3A — x — hos 3A. 
and since 


(cos s — cos 3A) = (vers 3A — vers 8) 
h cos 1A = (R — R’) vers 3A — (x — R’ vers s) (299) 
The last term is the expression for p, whence 
(R — R’) vers 3A = hceos 7A + Pp (300) 


and 
p = (R — R’) vers 34 — h cos 7A 


194 FIELD ENGINEERING 


From the figure, 
AS =SM —MC—CA 
d=y-—R’ sins — (R — R’ +A) sin 4A (301) 
or 
d=q—(R— R’ +A) sin 3A 


In view of the probable value of D’ select a suitable spiral, 
and with the tabular value of p and any desired small value 
of h solve eq. (300) for (R — R’) whence the approximate 
value of R’ is known. From the table of Radii correct 
R’ to the nearest minute, and with this value of R’ find h 
by eq. (299) and d by eq. (801). The last term of eq. (299) 
gives the correct value of p. Length, original, d + 4L; 
new with spiral, nc + 3L’. 

If desirable a comparison of lengths may be made with 
the approximate value of (R — R’) before finally fixing 
on R’. 

Example.—Given: D, 6° 00’; A, 50° 12’; select spiral 
n 9, c 26, s 7° 30’. 

Assume h 1 foot and p (from table) 3.120; then, approx. 
(R — Rk’) = 42.65; but adopt FR’ for 6° 16’; and find, eq. 
(299), hi=.0.991,. p = 2.988, ¢ = 114.179, d= 96.529. 
Lengths, original, 514.86; new, 514.85. 

226. Given: A simple curve, to replace it by another with 
spirals that shall pass through the same middle point, 
H. 

This is a special case of the last problem in which h equals 
zero. Select a spiral for D’ a little greater than D and use 
the tabular value of p in eq. (800) to find an approximate 
value of R’. Take the tabular value of R’ for the nearest 
minute and proceed as before to solve eq. (299) and eq. 
(301). This will result in a very small value of h, either 
plus or minus, which must be noted. 

Example—Given; D, 5°, A 48°, Spiral, n 7, c 26, p 1.59. 
Then, eq. (800), approx. (R — #’) = 18.39 and R’ 1127.89. 
Assume R’ = 1127.50 for 5°05’. Then p = 1.554,h = 0.077, 
q = 90.139, d = 82.469. 

Otherwise, to make h precisely zero, we have from the 
equation above eq. (299) 

R! ma vers gAsAir 


~ cos s — cos 3A (302) 


THE SPIRAL CURVE 195 


and use the corresponding fractional value of D’. Find 
d by eq. (801). 

227. Given: A simple curve joining tangents, to compound 

it near each end so as to introduce a spiral. Fig. 102. 

In place of 7A use some smaller angle 6, compounding at 

some point Q, and by substitution the last equation becomes 

, . & vers 06 — x 

; ~ COS Ss — Cos 6 303) 

Use the corresponding value of D’ for the are LQ. In eq. 
(301) substitute @ for 3A, and solve for d. 


A 





hig. LOZ: 


Otherwise, assume the nearest tabular value for R’ and 
with this compute a new value for 0, thus fixing the position 
of the point Q. For this purpose, in eq. (303) write (vers 
6 — vers s) for (cos s — ccs @) and solve for vers 6, whence, 
x — R’ vers s 

R.— R’ 

The numerator is evidently the new value of p. 

In eq. (301),make d equal zero and write 6 for 3A, whence 


d=y-—R’'sins — (R — R’) sin 8 (305) 


vers plz (304) 


196 FIELD ENGINEERING 


Example.—Given: D 3° and Spiral n 7, c 40, and @ = AOQ 
= 9° 10’. 

By eq. (303) 

R’ = 1729.31 or D’ = 3° 18’ 45”; but assuming D’ 3° 20’, 

by eq. (304) 
2.442 
190.96 
By eq. (3805) 

d = 279.802 — 139.865 — 30.442 = 109. 495 


Hence old line = SAQ = 415.255; new line = SLQ = 
415.190. 


vers 9 = = vers 9° 10’ 22”, and AQ = 305.76 





imhvel. OB. 


Note.—We may assume R’ without using eq. (303) and if 
6 be not satisfactory, try again. If we assume R’ from D, 
for (1 + 1) of the spiral, we may use the tabular value of 
p for the numerator in eq. (304). 

228. Given: A simple curve joining two tangents, to move 
the curve inward so as to provide clearance for 
spirals. Fig. 108. 

a. When the spirals are alike, the clearance is expressed by 
p in eq. (290). In eq. (800) replace (cos s — cos $A) by 
(vers 34 — vers s) and make I’ = Rk, whence 

ee aoe VETS Ss 0 
cos 3A cos 4A 
The minus sign indicates movement toward the center. 





(306) 


THE SPIRAL CURVE 197 . 


Similarly, for the distance AS we obtain from eqs. (301) 
and (291) changing the sign of h, 


d=q+hsin 3A =q+>p tania (307) 


The maximum displacement, h, occurs at the middle of the 
curve. 

Both eqs. (306) and (807) may be easily derived from the 
figure as in- § 225. 

b. When different spirals are required: Fig. 104. 





Fie, 104. 


With the given value of R compute P, p’, g, a, by eqs. 
(290) and (291), and let p be less than p’. 

In the figure let V’ be the intersection of paralie) tangents, 
WH =p and V’K = p’.. The diagonal VV’ = AA' = BB’: 


AC =VH =VF—HF BC’ =VK =VG— kG 


RC Gt Ey NET ey 
~-gin A tan A a sin A tan A (308) 


d=q+AC d’=q' + BC’ (309) 








198 FIELD ENGINEERING 


Note that if BC’ should prove negative in (308) it takes 
the minus sign in (309). The maximum displacement of 
the curve equals VV’ and is on the radius drawn parallel 
to VV’. 


ny _ AC algal 2 
tan VV’H = : pit VV’ = MRT, (310) 


Example.—Given: D = 5°; A = 37°; and spirals nc = 
85030; 1'C) =105X< 86: 





Fra.) 105; 


Then ) & ='6° 3. ="9° 10% 5p = 2.61 b ep ees, 
q = 119.903; 9’ = 176.427; AC =5.641; BC’ = 2.931; 
d = 125.544; d’ = 173.496; VV’H = 65° 08> VV’ = 6.218. 

The necessity for differing spirals seldom occurs with a 
simple curve. HH the displacement should be objection- 
able for any reason one of the preceding methods may be 
adopted instead. . 


Spirals with Compound Curves 


229. Given: A Compound Curve joining tangents, to 
move the entire curve inward so as to provide 
clearance for spirals. Fig. 105. 


THE SPIRAL CURVE 199 


As suitable spirals will necessarily differ, the general method 
of the last. section will apply here. Designate the spiral 
values by the same subscripts as the ares themselves, let 
R; be less than R, and A = A; + Ax. With R, and gs, find 
pi and qm; also with fz and s: find p: and q@, using eqs. 
(290) and (291). 

Then, as in eq. (308): 


AC me El Pe BO = P2 Pi 


sin A tan A sinA  tandA 








(311) 


If BC’ should prove negative give it the minus sign in the 
next formula. 


d=u+BC ° dz = @+ AC 
yagee as PEA stl SCAM 
tan VV = AG VV cos VW 


The maximum displacement of the curve equals VV’ 
and occurs on the radial line drawn parallel to VV’. If 
this radial line does not cut the curve between tangent 
points the curve displacement varies between the limits p; 
and pz. The central angles of the resulting curve will be 
respectively 

(Ar — si) and (Az — 82) 

Example.—Given: 


dD, 8° 20°: Ay 25), 03’, Ny1C1 11 ee 24, Si 12 
Dz 3° 20’, Ae 38° SU, neec2 8 X 45, S82 6° 


Then we derive 


pi 4.974, pz 3.923, qi 131.680, ge 179.883 
AC 3.621, BC’ 1.938, d; 135.301, d, 181.121 
VV’H 42° 42’, VV’ 5.338 max. at 17° 39’ beyond P. 


230. Given: A Compound curve joining tangents, to re- 
place it by another with spirals on the same ground and 
of the same length, as nearly as may be. Fig. 106. 

The point of compound, P, will be preserved, but moved 
outward by a small offset, h, and each arc may then be 
treated separately as though it were half of a simple curve. 
As each are will be somewhat sharpened, select suitable 
spirals accordingly, assume a small value for h (see Table 
XL for values of h in various cases) and compute the new 
radius R,’ and also R,’, giving probably fractional degrees of 


200 FIELD ENGINEERING 


curve. These may be adopted at once, or, if preferred, one 
new radius may be assumed for the nearest minute, and 
with it a new value for h computed, and with this recom- 
pute the value of the other new radius. 

With these final values proceed to compute d; and dy, 
and compare the lengths of the new and old lines. The 





Fia. 106. 


formulas for one are are the same as in § 225 with mere 
change of notation; and are as follows: 


a R vers A, — (x +h cos A)) 


. 


a cos s — cos Ay SA 
h cos A = (R — Ry’) vers A; — (x — Ry’ vers s) (313) 
d=y—R,' sins —(R—R,’ +h) sn A (314) 


The same formulas apply to the other arc, only changing 
the subscripts where they occur. 


THE SPIRAL CURVE 201 


Example.—Given: 


eB; 429° 7 9c 20 
and 
D2 6°; Az 25° 06’; n 9, c 26 


Assume h=1. Then D,’ = 6° 16’ 06”; but for D,! 
BLOT ast = .99054, and for this value of h, D,’ = 8°17’ 56” 
(say) 8° 18’. Then d; = 76.474, d, = 96.981. 





Fia. 107. 


Length old line 439 + 515.3 = 954.3 
Length new line 439 + 514.9 = 953.9 


Given: A Compound curve, to apply spirals without dis- 
turbing the middle portion of the curve. 

The methods discussed in §§ 226 and 227 apply here 
with merely a change in notation; letting P replace H, 
and A; or A; or.@ replace $A as the case may require. In the 
case of three or more ares in the curve only the terminal 
ares are affected by the use of these methods. 


202 FIELD ENGINEERING 


231. Given: A Compound curve, to introduce a_ spiral 
between arcs at the point of compound, P. Fig. 107. 

The ares must be separated by a radial offset at P to 
afford clearance for the spiral, the sharper arc being inside 
of the other. In general the spiral and offset bisect each 
other. In Table XXXVI under any one chord-length find 
two values of Ds corresponding, as nearly as possible to D, 
and D, and use the portion of the spiral included between 
them; lay off one half this length on each are from P, for the 
new compounding points, F and G. From the point F the 
spiral FG may be located by deflections from the local 
tangent, using only the portion of spiral selected, and the de- 
flections given below the zeros in Table XX XV for the point 
number represented by F. 

Or the spiral may be located from H towards F by using 
the deflections given above the zeros in Table XX XV for the 
instrument at the point number represented by H. The 
sum of the angles taken from the two arcs equals the spiral 
angle for the portion of the spiral used. Should the dis- 
‘tance FH prove to be a little longer or shorter than the 
spiral the several chord-lengths may be slightly altered. 

Now the linear offsets from are to spiral at the chord 
points are the same as from a tangent, regarding F or H as 
the point of Spiral S, and numbering the points from them 
as usual; the offset is equal to x given in the table. Thus, 
if the spiral is eight chords long, the fourth point from either 
end will be at P, consequently the offset PP’ will equal 
2x for n = 4 in the proper table. 

But when the spiral has an odd number of chords the 
half length will bisect the middle chord, and the value of z 
is found by multiplying the chord by the coefficient here 
given opposite the half length of spiral. : 


Coefficients for x at Half-chords 


n n n 
1.5 .00364 4.5 .05999 7.5 .24711 
7 eo ALES 5.5 .10397 8.5 .35200 
3.5 .03054 6.5 .16538 9.5 .48288 


The spiral may be located by offsets, first laying out 
chord-lengths from P on one arc, and from P’ on the other, 
and setting off x at each point till F or H is reached, where 


THE SPIRAL CURVE 203 


x =0. If the spiral long chord FH be required, it may be 
computed by eq. (288) for the portion of the spiral actually 
used. 

Example.—Given: a compound curve of D 4° and D’ 12° 
to introduce a spiral at the P.C.C. In Table XXXVI under 
c 21 find at n 5, Ds 3° 58’ and at n 15, Ds 11° 55’; therefore 
the spiral will extend from point 5 to point 14, and the spiral 
angle s will be (17° 30’ — 2° 30’) = 15°. Divide this 
between the arcs in the ratio of their degree of curve, or. 
; and ? parts, 3° 45’ on the 4° curve, and 11° 15’ on the 
other. This takes 93.75 feet on each arc, total 187.50 feet; 
but the spiral measures 9 X 21 = 189 feet; the difference, 
1.50 feet divided by 9 gives a correction of 0.17 for each 
. chord-length; hence ¢ = 20.83. The point n 5 is taken 
at F, and from Table XX XV ‘“ Inst. at 5” the deflection for 
point 6 is 30’. Make this and the succeeding deflections 
up ton 14orH. At H deflect from chord HF to find tangent 
8° 36’ 45” (found “ Inst. at 14” n 5) to complete the 15°. 

The middle point of spiral is n 4.5 for which the coefficient 
is 0.06;- hence the clearance p = 2 X 21 X 0.06 = 2.52 
for separation of curves at P. 


Field and Office Work 


232. A spiral may be set out upon the ground in either 
direction by using the proper list of deflections, according 
to the point n at which the transit is placed, but to secure 
perfect agreement with the tangent it is better to proceed 
from the tangent point with each spiral, and afterward 
connect them by the intervening curve. The several points 
of the spiral, however, are not required except for track- 
laying, and in the earlier stages of the work it is sufficient 
to locate the extremities of the spiral by its long chord, 
SL. Any regular station of the survey which falls upon the 
spiral may be located by treating it as a plus between chord- 
points, making the deflection to it acccrdingly by ordinary 
interpolation. Strictly speaking, there is a second differ- 
ence to be applied to the deflection so found (minus when 
running from S and plus when running from L), but as this 
amounts only to 12’’.5 at the middle of a chord, it may 
usually be ignored. A transit point, if required on a spiral, 
will be located at a regular chord-point. 


204 FIELD ENGINEERING 


In a difficult country, where the line requires close adjust- 
ment to the contour, it is wise to ignore all spirals until a 
satisfactory location has been obtained. Spirals may then 
be introduced by any of the methods described. Only in 
a flat country, or where a very careful paper location has 
been projected, is it worth while to set out spirals at the 
first running. 

In projecting a paper location it is desirable to have at 
hand a set of circular templets cut to the scale of the map 
and representing convenient degrees of curve. After find- 
ing by trial a curve that fits the topography when touching 
the tangents, locate the tangent point A by a perpendicular 
from the center, and withdraw the templet a little to give 
the necessary clearance p for the spiral. The distance moved 
along the line VO is 

P 

~ cos 2A se 
The clearance may be too small to be shown accurately to 
scale, but the distance AS may be computed, eq. (307), 
and laid off to locate S from which the point L may be 
located by its coordinates, x and y. Or, the entire tangent 
distance VS may be computed and scaled. The spiral may 
be drawn with a suitable curved rule. It should sensibly 
bisect the offset p and be bisected by it very nearly. 

When curves predominate these should be drawn first 
to suit the contours, but far enough apart to admit the 
necessary spirals and a bit of tangent. The tangent lines 
are then drawn to touch short ares outside the curves at the 
distance p. Then froma radius drawn perpendicular to 
the tangent lay off q to locate S, and R sin s to locate M 
opposite L. 

Table XX XIX supplies the quantities required for project-- 
ing curves and spirals in either field or office work for a selected 
list of simple curves. It furnishes also a means of computing 
correctly the spiral tangent 7's for any curve in the list by 
the use of the formula 


T; =T +q+ p tan $A (316) 


in which T is the tangent distance for the simple curve. Much 
time may be saved by adopting in practice the curves listed 
in this table. 


CHAPTER X 
LEVELING 


233. The field operations with the Engineer’s Level 
are of a more simple character than those performed with 
the transit, yet require equal skill and nicety of manipula- 
tion in order to produce trustworthy results. The transit 
- is used to ascertain the relative horizontal position of points, 
the levél to obtain their relative vertical position. 

234. In order to express the elevation of points, they 
must be referred to some level surface of known (or assumed) 
elevation; and in order that the elevations may all be posi- 
tive upward, this surface of reference should be selected 
below all the points to be considered. The level surface 
of reference is called the datum. 

The elevation of the datum is always zero. The elevation 
of any point ts its vertical height above the datum. 

Near the coast the sea level is usually adopted as the 
- datum; inland, the low water mark of a river or lake, etc.; 
but it is not necessary that the datum should coincide with a 
water surface. If any points whose elevations are to be 
ascertained are below the water surface, the latter may be 
assumed to have an elevation of 100 or 1000 feet instead. of 
zero; that is, we remove the datum, in imagination, to 
109 or 1000 feet below the level of the water surface. 

235. In case of a survey commencing at a point quite 
remote from any important water surface, any permanent 
point may be selected as the original point of reference, 
and its elevation may be assumed at 100 or any other number 
of feet; that is, we fix the datum at the same number of feet 
below that point. The point of reference is called a bench, 
or benchmark, and is designated by the initials B.M. 
Other benches are established at intervals during a survey, 
and their elevations determined instrumentally. They 


205 


206 FIELD ENGINEERING 


are then convenient points of known elevation for future 
reference. We cannot asswme the elevation of more than 
one bench on the same survey, else we should have more 
than one datum, and all the results would be thrown into 
confusion. 

236. Having established the first bench and recorded its 
elevation, the next step is to set.up the instrument firmly 
at a moderate distance from the bench, so that the telescope 
shall be somewhat higher than the bench, and in full view 
of a rod held vertically upon it. The instrument having 
been tested for its several adjustments, and found to be 
correct, the line of sight through the intersection of the cross- 
hairs is known to be horizontal when the bubble stands 
at the middle of its tube. Turning the line of sight upon 
the rod, the point of the rod covered by the horizontal 
cross-hair is known to be on a level with the cross-hair; 
and the latter is therefore higher than the bench by the dis- 
tance intercepted on the rod from its lower end. Adding 
this distance to the elevation of the bench, we obtain the 
elevation of the cross-hair, known technically as the ** Height 
of Instrument,’’ and designated by the initials H.1. 

237. The distance intercepted on a rod from its lower 
end by the line of sight, when the rod is held vertically on 
any given point, is called the reading of the rod at that point. 

238. Having obtained the height of instrument, the 
elevation of any point somewhat lower than the cross-hair 
is easily ascertained by taking a reading of the rod upon it. 
The reading subtracted from the height of instrument 
gives the elevation of the point above the datum. The 
elevation of any number of other points may be similarly 
obtained. But the elevation of points on the ground higher 
than the cross-hair, or farther below it than the length of 
the rod, cannot be determined, because in either case the 
line of sight will not cut the rod, and hence there can be no 
reading. In order to observe such points, the instrument 
must be removed to a new position, higher or lower than 
before, as the case may require. 

239. Before the instrument is removed to a new posi- 
tion, a temporary bench, called a Turning Point (and 
designated by T.P. or “ Peg’’) must be established, and its . 
elevation ascertained as for any other point, but with more 


LEVELING 207 


care. A turning point must be a firm and definite point 
whose position cannot readily be altered in the least, nor 
lost sight of. A small stake firmly driven, or a point of rock 
projecting upward, is frequently used. The reading having 
been taken on the turning point, the instrument is carried 
forward to a new position, leveled up properly, and the new 
Height of Instrument obtained by a new reading on the 
same turning point. Since the cross-hair is higher than the 
point (otherwise there could be no reading) the reading, 
added to the elevation of the point, gives the Height of 
Instrument. 

240. In general, the intersection of the cross-hairs being 
higher than any point on which a reading is taken: 

_ To find the Height of Instrument, add. the reading on a point 
to the elevation of the point; and 

To find the Elevation of a point, subtract the reading on 
it from the Height of Instrument. 

A reading taken for the purpose of finding the Height of 
Instrument is called a Backsight (B.S.). A reading taken 
for the purpose of finding the elevation of a turning point 
(or of a bench used as such) is called a Foresight (F.S.). 
Hence Backsights are always plus, and Foresights always 
MINUS. 

It is to be noted that the terms Backsight and Foresight 
- do not refer to the directions in which the sights are taken. 

241. The form of field-book used for the survey of 
a railroad, or other continuous line, is shown below. The 
first column contains the numbers of the stations on the line 
and of plus distances to other points on the line where read- 
ings are taken—also the initials of benches and turning 
points, in order, as they occur. The second column contains 
the backsights, taken on points of known elevation only. 
The third column contains the height of instrument, recorded 
on the same line as the elevation of the turning point (or 
bench) from which it is calculated. The fourth column 
contains the foresights, taken on new turning points, and 
benches used as such, only. The fifth column contains 
the readings taken on al other points noted in the first 
column. The sixth column contains the elevations of all 
points observed. The right-hand page is reserved for 
remarks, descriptive of the benches and their location—of 


208 


objects crossed by the line, as roads, streams, swamps, ditches, 


FIELD ENGINEERING 


etc., the depths of streams, etc. 


LEVEL BOOK. 








Sta. B.S fs bal be F.S Rod. Elev Remarks. 
Ba Vite 4 OS smi CUS = OSduiictars coskae ktecudee 200.000 | White oak, 115 R. 
1) ROP SRE ES EEE eee eS ahe eke Pee | 202.6 
a eee lee a Ne Cee 3.4 201.3 
cc 15) Oi eae RI cue cee A =e nage aan hl ne 199.5 
TP 1.791 | 197.260 Ont Week 195.469 
PU CR dards Ask! hevad Ge ae ee 3h ft 193.6 
at HU Ae rae, eRe eee 70 190.3 Brook 5 wide; 1 deep 
te HOL | Seiad «laste ae All eteeeenene Baal 194.2 
Recto) faa Gm SANE cites 2 SERIE eas 9h es 0:5 196.8 
TP L170 ae 208; 574 074860 196.824 
T.P. | 11.988 | 219.528 O97 OR ee, oe 207.595 
OD eee a cre tel Dans cok ee ee Ae 3.5 216.0 
A feb spo’ asd xe heals. acces | ees % 2.6 216.9 
1 BYCGTNNY 71 MA dil Wim inl ce 25 a. 2.075 | 217.453 |Maple, 78L. 
FAG Woes ado See RES Roll aac eae 137 217.8 
Cee SPS etamaer ai eas e  e  Se 0.9 218.6 
ee 9.005 | 227.801 ORISO) ae ee 218.796 
oll es tO Se oes vane NY eal te ee 221.6 
39.162 11.361 




















When a bench is not used as a turning point, the reading 
on it is recorded in the fifth column. 

The numbers in the second, fourth, and fifth columns 
come directly from the rod, those in the third are obtained 
by addition, those in the sixth by subtraction, according to 
the rule given above. The additions and subtractions 
made on each page should be proved before proceeding to 
the calculations of the next. When correct, the difference 
of the sums of the hacksights and foresights on the page 
equals the difference of the first and last elevations on the 
page. Thus, in the form given 


(39,162 — 11.361). = (227.801 — 200.000) = 27.801 


In this proof we ignore all elevations except those of turn- 
ing points, and benches used as such, and the height of in- 
strument. 

At the end of the survey, as well as at the end of each 
day’s work, a bench is established from which the survey 
may be resumed at any future time. See §§ 28, 29 and 80. 

242. The object of making such a survey with level and 
rod is to furnish a_ profile or vertical section of the entire 
line, showing in detail the rise and fall of the surface over 


LEVELING 209 


which it passes. The profile is plotted on profile-paper 
published for the purpose, the horizontal scale being usually 
400 feet to an inch, and the vertical scale 30 feet to an inch. 
This distortion of scale magnifies the vertical measures so 
that slight changes in the elevation of the surface may be 
seen distinctly. 

243. When only the difference of level of two extreme 
points is required, the survey is more simple. No readings 
are taken except on turning points, the backsights and 
foresights being recorded in separate columns. No cal- 
culation is required until the survey is finished, when—the 
first reading having been taken on one of the given points, 
and the last on the other—the difference of the sums of the 
. backsights and foresights is the difference in elevation of 
the two points, according to the method of proof mentioned 
in § 241. ‘Thus the difference in level of any two benches 
established on a previous survey may be tested, and, if found 
correct, all the intermediate elevations on the line may be 
assumed to be correct also. The discrepancy should not 
exceed one tenth of a foot in any case, and is usually much 
less. 

244. Any lack of adjustment in the instrument gives 
the line of sight a slight angle of elevation or depression 
causing a slight error in every reading, proportional to the 


~ distance of the rod from the instrument. But the errors 


being equal for equal distances, and the backsights and fore- 
sights having opposite signs in our calculations, the errors 
cancel when the distances are equal. Hence, to avoid 
errors in elevation, each new turning point should be as nearly 
as possible at the same distance from the instrument as the 
point on which the last backsight was taken. For precise 
reading, the rod should not be more than 400 feet from the 
instrument. 

245. Another cause of error in readings is want of verti- 
eality in the rod. This may be avoided by the use of a disk- 
level, or in the absence of wind, by balancing the rod. The 
rod may be plumbed one way by the vertical cross-hair 
of the level, and to ensure a vertical reading in the plane 
of the line of sight, the rod may be gently waved each side 
of the vertical toward and from the instrument, the shortest 
reading being the correct one; or in case of a target rod, 


210 FIELD ENGINEERING 


the target should rise to, but not above the horizontal cross- 
hair, as the rod is waved. 

246. When very long sights are required to be taken 
with the level, another source of error must be considered, 
namely, the curvature of the earth. 

A level line is parallel to a great circle of the earth, and is 
therefore an are of a circle, or may be so considered. 

A horizontal line is a straight line parallel to the plane 
of the horizon. Therefore the line of sight, being a hori- 
zontal line, is tangent to the circle of a level line passing 
through the instrument. 





Fia. 108. 


To find the correction in elevation due to curvature of the 
earth for any distant station. Fig. 108. 

Let A be the station of the instrument J, and B the dis- 
tant station observed. 

Let Ro = CI = the radius of curvature of the earth, or 
of the parallel are JD. Let L,) = ID = the level distance 
between A and &. Then JE, perpendicular to CJ, is the 
line of sight, BE is the reading of the rod, and DE = Ey = 
the correction due to curvature. 

By Table XLI, 24, [H? = DE(DE 4 2Ro); but since 
DE is very small cbt pahen with 2Ro, it may be omitted from 
the parenthesis, and since JH =ID = Lo very nearly, 
because the angle ACB is very small, we have Lo? = 2RoL. 
= Lo? 
Ko = aR, 


Eo is to be added to the apparent elevation of station B. 


(317) 


LEVELING ati 


247. Refraction. In observing distant stations the 
line of sight passing through the atmosphere is refracted 
from the straight line JZ, Fig. 108, and takes the form of 
a curve, which, for practical purposes, may be considered 
as the are of a circle, concave downwards. Its radius, de- 
pending on the conditions of the atmosphere, varies from 
53 to 74 times the radius of curvature of the earth. 7Ro 
is considered a good average value. 

Refraction causes the observed object to appear too high, 
while the curvature of the earth causes it to appear too low . 
the effects being contrary, the correction for curvature is 
reduced by the correction for refraction. If we let Ho = the 
total correction for both curvature and refraction, to be added 
_ to the apparent elevation of the observed object, then 
3 6 3 Li? 
vine aon e 
Table XVII is calculated by this formula, assuming a mean 
value of Ry = 20,913,650 feet. 

248. The form of the earth is approximately an ellip- 
soid of revolution. Its meridian section at the mean level 
of the sea is an ellipse, the semi-axes of which are, according 
to Clarke, 

at the equator A = 6,378,206 meters [6.8046985] 
atthe poles 5B = 6,356,584 meters [6.8032238] 
According to the same authority 


Ho = (318) 


1 meter = 3.280869 feet [0.5159889] 
Therefore the semi-axes expressed in feet are 
A = 20,926,058 feet | [7.3206874] 
B = 20,855,119 feet [7.3192127] 


Then the radius of curvature of the meridian 


2 
at the equator, se = Ry = 20,784,422 feet [7.3177379] 


2 é 
at the poles, - = Ry = 20,997,240 feet [7.38221622] 


In latitude 40° the radius of curvature of the meridian is 
20,871,900, and of a section at right angles to the meridian, 
20,955,400; the mean value, or Ro = 20,913,650 [7.320430], 
being adopted for general use. The error in the correction 
Ho eq. (318) due to this assumption will usually be much 


2tY FIELD ENGINEERING 


less than that due to the assumed value of the radius of 
refraction. 

249. Leveling by Transit. When a transit has a level- 
tube attached to the telescope, it may be used for finding 
the difference in elevation of points. If the instrument 
be in perfect adjustment, the line of sight will be horizontal 
when the bubble stands at the middle point of the tube, 
and the reading of the vertical circle will be zero. Should 
there be a small reading when the line of sight is horizontal 
it is called the index error. When the line of sight is not 
horizontal, the angle which it makes with the plane of the 
horizon is called an angle of elevation, or of depression, 
according as the object upon which the line of sight is 
directed is above or below the telescope. This angle is 
measured on the vertical circle, being the difference of the 
reading and the index error, when both are on the same side 
of the zero mark, and their swum, when they are on opposite 
sides. When the distance to an observed object is known, 
and its angle of elevation or depression is measured, we may 
calculate its vertical height above or below the telescope. 

{ elevation; 
\ depression; 
L = the horizontal distance; 
L’ = the distance parallel to line of sight; 
h = difference in elevation of object and instrument. 
Then for short distances, 
h=Ltane = L’' sine (319) 

For long distances the curvature of the earth and refrac- 
tion must be considered. Fig. 109. ; 

Let J be the place of the instrument, and F the object 
observed. : 

Let Lo = the distance, measured on the chord of the 
level are JD, passing through the instrument; and let y = 
the number of seconds in the are 1D; hence, since for ordi- 
nary distances the chord and arc are sensibly equal 


py = / 2062648 [5.314425] 
0 


Let +a = angle of 


or giving to Ro its mean value, § 248, 
y = Lo X .0098627 [7.993995] 


or a fraction less than 1’ per 100 feet. 


LEVELING 213 


Let JF be the are of the refracted ray, and assuming 
that its radius is 7Ro, the are will contain one seventh the 
number of seconds of the are JF. 

IF’, tangent to JF, is the direction of the telescope; JF 
is the chord of the are JF, and JE is the horizontal. 

Let a= EIF’ = observed angle of elevation. Then 
EIF = true angle of elevation = EJF’ — F'IF =a — }-4y 
=a — 0.071y. 

The angle EID =}y .». DIF =4¥ +a —0.071y; and 
IDF = 90° + 3y. «. IFD = 90° — (¥ +a — 0.071y). 





C j 
Fig. 109. Fie. 110. 


We now solve the triangle 7FD for the side DF = h, 
and find 
sin (4¥ +a — .071y) 
cos (Wy +a — O71Yy) 


For an observed angle of depression make a negative in 
the formula. - 

The coefficient 0.071 is called the coefficient of refraction, 
this being a fair average value, while its extreme range is 
from 0.067 to 0.100 under varying conditions of the atmos- 
phere, and values of the angle a. 

When the difference in elevation of two or more distant 
objects is required, we obtain the elevation of each sepa- 
rately, and subtract one elevation from another. The 
elevation of the observed object is given by (H.I.) +h. 

250. To find the Height of Instrument of a transit 
by an observation of the horizon. lig. 110. 


h== Lie 





(320) 


ag Vs eel FIELD ENGINEERING 


Let I be the place of the instrument, and let a = observed 
angle of depression of the horizon. 

Let F be the point where the refracted ray meets the 
level surface, and draw the chords JF and AF. 

Let y = the angle ACF, let h = AJ, and let k = the 
coefficient of refraction. 

In the triangle JAF, 
IAF = 90° + 4y, AFI = 4y —ky, AIF = 90° — (y — ky) 

Hence FJE =y —ky. But FIE =a+ky 

a 
ae pony 

Let Ff” be the tangent point of a right line drawn through 
I; then AJ = CF” ex sec ACF”, but CF” = Ro, and, since 
y is always very small, ACF” = }(y +a) very fe 

1—k 
% LV 2k 





(321) 





1-—k 
1 — 2k 
Giving to Ro its mean value, § 248, and assuming k = 7y 

log h = 7.320430 + log ex sec. 1.0801 a (323) 


Otherwise; we may solve the triangle A/F since 


h= Rp ex sec a (322 














Bie Aas ees ; a 
AF = 2Ro sin $y = 2Rp sin a — 2h) 
and 
a sin (gy — ky) 
ie Ad Ob (y — ky) 
+. : a / sin 5a 
hg B io 81M oer ony : dik (324) 
OST 9 
When k = x4 
ee OR cine (325) 


OSs 13a . 
_Lxample.—The observed dip of the sea horizon is 24’ = a. 
What is the height of the instrument above the sea? 
By eq. (823) 1.0801 X a X 60 = 1555.34 3.191825 
2 


6.383650 


Table XXVI (q — 2) 9.070130 
Ro 7.320430 


-7 h = 594.58 2.774210 


LEVELING ANS 


Methods of determining heights by distant observations 
cannot be relied on for more than approximate results, since 
they necessarily involve the uncertain element of refrac- 
tion, and usually a lack of precision in the vertical angle, 
the arc reading only to minutes in ordinary instruments. 
These methods, however, are useful where no great accuracy 
is required, as for a temporary purpose until levels can be 
taken in the regular way, or for interpolating between points 
of established elevation. 





Fra. 111. 


251. Stadia Measurements. It is sometimes conven- 
ient to determine distances by instrumental observation. 
For this purpose two additional cross-hairs may be placed 
in the telescope parallel to each other and equidistant from 
the central cross-hair. These are called stadia hairs, and 
distances determined by them are called stadia measure- 
ments. The stadia hairs are adjusted so as to intercept 
a certain space on a rod held at a certain distance from 
the instrument and perpendicular to the line of sight. For 
any other place of the rod, the distances and intercepted 
spaces are nearly proportional. The exact relation is given 
below. Fig. 111. 

Let M and.N be the position of the stadia hairs, the dis- 
tance between them being denoted by 7; s = CD = the 
space intercept on the rod between the two hairs. 


216 FIELD ENGINEERING 


The triangles MON and COD are similar, and 
fy ey eC LIOR, 
fo 8 


ae (326) 


Let f be the focal length of the objective lens; then from 
Optics, we have 


or 


eens vor 


| hee ae 
which, combined with eq. (826) will give 
fr = f, eres 
Now fo + ¢ = the distance from the center of the instru- 
ment to the rod = D 


=(p+o=Fs+(fto=kst+(f+e) (827) 
If the rod is held at any other distance from the instru- 
ment, its position may be found by multiplying the space 


intercept on the rod by Land adding (f +c) to the product. 


The values of f, c and £ may be furnished by the maker, 


but if not, they may be determined by the observer. The 
value of f is found by focusing the object glass on a point 
at an infinite distance, and measuring the distance from the 
plane of the cross-wires to the center of the object glass. 
The value of c is the distance from the center of the instru- 
ment to the center of the object glass when focused at a 


point about 200 feet away. To find = i lay off a distance 


(f +c) from the center of the ee which locates a 
point below F in Fig. 111. Take intercepts at different 
distanees from this point. Divide each distance by its 
intercept. The average of the different quotients will be 


the value of = ! or k. 


It is to be sefie that the work should be done on a nearly 
level stretch of ground and that the telescope bubble should 
be practically in the center at all sights; that is, in order to 
effect a coincidence the upper or lower cross-hair should 


LEVELING 217 


not be moved over more than the smallest’ graduation of the 
rod. 

252. Inclined Sights. Fig. 112. If the rod were held 
perpendicular to the line of sight, a portion of it would be 
in the position C’D’. It is not practicable to do this, as there 
is no easy method of securing the proper angle of inclina- 
tion; therefore, the rod is held vertical, and a correction is 
made for this position. The inclined distance, /G,is then — 
obtained, and finally the horizontal distance JE, 





Hie, £12; 


Let: G’D! =,s' and .CD = s. 
Then C’D’ = CD cosa = s cosa 
Now 


IG =ks’ + (f +c) =ks cosat+(f +c) 
But 

TH =H =1G cosa 
and substituting the value of JG, we get 

H =ks cos? a+ (f +c) cosa (328) 

Similarly, 

GE =V =1G sina 
or 

V=kssmacosa+(f+c) sina 


= tks sin 2a+(f +c) sina (329) 


Reductions:—Table XVIII is used in finding horizontal 
distances and differences in elevation, 





218 FIELD ENGINEERNIG 


Under “ Correction to Horizontal Distances” are given 
100 minus ks cos? a for different values of a, and for k = 100 
and s = 1. Proper multiplication is made for other values 
of these quantities. The value (f + c) cosa is found at the 
foot of the page for three values of (f + c). 

The quantities in the table under “ Diff. Elev.”’ are values 
of ks sin 2a where k = 100, s = 1 and a = the particular 
vertical angle. With any other values of k and s the prod- 
uct is found by use of the slide rule. The quantities at the 
foot of the column are values of (f + c) sin a for three values 
of (f +c). The products of the other values of (f +c) are 
found by interpolating. 

Ezxample.—Angile 2° 46’; rod reading 2.64; f +c = 1.1; 
and k = 100. 

Difference in elevation = 4.82 < 2.64 + 0.04 = 12.76. 

Correction for horizontal distance for 100 feet = 0.23 
and for a rod reading of 2.46, it is 2.46 < 0.23 or 0.6. Cor- 
rection for (f +c) cos a is 1.1. Therefore the horizontal 
distance = 246 — 0.6 + 1.1 = 246.5, 


CHAPTER XI 
CROSS-SECTIONS 


253. The established grade upon the final profile of a 
located line is the basis to which all the work of construction 
must conform. The roadbed must be brought to grade and 
made of a prescribed width, while its slopes, either in cut 
_ or fill, must be provided for. Before the contractor can pro- 
ceed a-portion of the work must be staked out for him and a 
record of the measurements made by which the amount 
of material to be moved may be computed. This staking 
out of earthwork is called cross-sectioning. All cross-sec- 
tions are taken in vertical planes at right angles to the center 
line. } 

The frequency with which cross-sections should be taken 
depends entirely upon the form of the surface; where this 
is regular, a section at each station is sufficient. A cross- 
section should be taken, not only at every point on the center 
line where there is an angle in the profile, but also wherever 
an angle would be found in the profile of a line joining a 
series of slope stakes on either side, even though the profile 
of the center line may be quite regular at the corresponding 
point—the object being, not only to indicate the proper 
outlines of the earthwork, but to furnish the data neces- 
sary to calculate correctly the quantities of material removed. 
Rockwork will generally require more frequent sections 
than earthwork- | 

The base of a cross-section is identical with the width of 
the roadbed. It is made wider in cuts than in fills to allow 
for the side ditches. Six feet should be allowed in earth, 
and 4 feet in rock cuts. The ratio of the side slopes 
depends upon the material. The usual slope ratio for earth 
is 13 horizontal to 1 vertical for both excavation and em- 
bankment. Damp clay and solid gravel beds will stand 

219 


220 FIELD ENGINEERING 


for a time in cuts at 1 to 1, or an angle of 45°, but this cannot 
be permanently depended on. On the other hand, fine 
sand and very wet clay may require slopes of 1{ to 1 or 2 
to 1. Exceptional cases require slopes of 3 or 4 to 1. In 
rockwork the slopes are usually made at } to 1 for solid, 
z to 1 for loose, and 1 to 1 for very loose rock, liable to dis- 
integrate. Rock embankments stand at 1 to 1. 





Fie, 113. 


254. Since all points of importance on the natural sur- 
face have to be referred to grade to ascertain the depth of 
cut or fill necessary at each point, it is convenient to employ 
what is known as the Grade rod. ‘This is simply the differ- 
ence between any given height of instrument (H.I.) and the 
elevation of grade at the given point, or what a rod would 
read from the instrument if its foot were at grade. When 





Fie, 114. 


the grade is above the H.I. the rod is supposed to be inverted. 
The cut or fill at the given point is determined by comparing 
the rod reading at that point with the grade rod. Any change 
in the H.I. requires the finding of a new grade rod. With 
the same H.I., the grade rod changes with the grade from 
station to station. If rg be the grade rod and r any rod 
reading, then (Fig. 113) 


CROSS-SECTIONS apn 


Depth of cut 

Depth of fill 
but (Fig. 114) 

Depth of fill = rg +r when grade is above H.I. | 


Toss-iT 
Pang 


I 


(330) 


255. Cross-sections are divided into four classes, depend- 
ing upon the slope and shape of the natural surface, viz., 
level, three-level, irregular and side-hill. The following 
notation applies in any case (Figs, 115, 116, 117): 





Let 6 = AB, the base of the section or roadbed; 
a= gue = the slo 103 
Sr Di pe ratio; 
d = CG = the cut (or fill) at the center stake; 
h = DH = the cut (or fill) on the right of the section; 


k = EN that on the left; 
x =CD or CE the “ distance out’”’ from center to 
side stake. 


and 


D 









t 
| 
y | 
ee INS Mg ak eS ee ade 


Fie, 116: 


First read the rod at the station and find d from the grade 
rod, § 254; record d in the notes as cut or fill, and mark 
the same on the back of the stake; then proceed to locate 
other required points of the section, recording their distance 
from the center line and height or depth above or below 

grade. Drive stakes at the extreme points right or left, 


De FIELD ENGINEERING 


called slope stakes, and mark on each the proper cut or fill, 
facing the stake toward the center. Cuts are marked 
+ or C, fills — or F. 

Level section. Fig. 115. Sinceh =k =d 


x = 3b + sd (331) 
Three-level section. Fig. 116. 

x =4b+ sh } 

x = 3b -+ sk (332) 


Having found the center depth d, estimate from this and 
the apparent slope of the ground the probable depth h at 
the side, and from this compute a value of x (approx.) by 
eq. (3382), meagure out this distance to a point and take a 
rod reading which in eq. (330) gives a new h, and this in eq. 
(332) gives a new x; measure this cut as before, and repeat 
until the computed « agrees with the last measurement, 





ete led 


proving that the correct point for the slope stake has been 
found. This method applies to all cases of either cut or fill 
by observing to make the proper additions or subtractions, 
such as the case in hand requires. A little practice enables 
one to combine somewhat these several approximations and 
so to arrive more quickly at the required point. 

Irregular Section. Fig. 117. Proceed as in the last case, - 
but before seeking the extreme points find the distance from 
the center line to any point where the surface of the sec- 
tion changes, and record the depth to grade at such ‘point. 
Finally, by estimate and trial, locate the slope stakes as in 
the former case. 

In all three cases the method is the same whether the sec- 
tion is for cut or fill. 

Side-hill Section. A section is so named when its base 
or roadbed is partly in cut and partly in fill. Set the rod 
at the grade-rod reading and move the rod along the section 


CROSS-SECTIONS 223 


until the grade point is found; note its distance from the 
center and set a stake there, marked 0.0, facing it down 
hill. Find and note other points as in the other cases, but 
observe that 3b has different values in cut and fill, also that s 
may have two values, one for cut and one for fill. When 
the cut on the lower side is less than 1 foot is it well to go 
beyond that stake and set a grade stake, noting the distance 
out; thus.changing a thorough cut into a side-hill section. — 

256. When two materials are found in the same section, 
as rock overlaid with earth, each material requires its own 
slope, and a compound section is the result. To stake 
out work of this description, the depth of the earth to the 








Fie, 118, 


rock must be known, and may be nearly ascertained by 
reference to an adjacent section already excavated. Fig. 
118. 
Let a; be the depth of earth at C; . 
a, be the depth of earth at P or Q; 
s; be the ratio of rock slope; 
s, be the ratio of earth slope. 


Then x= jb4+5(d —a +y1) + Se(a2 + y) (333) 


in which y: = difference of rod readings on the rock at C; 
and D,, or C, and #,; and y = difference of rod readings 
on the surface at P and Ds, or at Q and HE». The upper sign 
applies to the upper side, the lower sign to the lower. 

It is better, however, to make an indefinite cross-profile 
at first, driving two reference stakes quite beyond the sec- 
tion limits; and when the contractor has removed the earth 
from between D,; and £,, indicate to him those exact points 


224 FIELD ENGINEERING 


by marks on the rock, and also set the slope stakes at Dz 
and EF). 
' 257. Terminal Sections. A grade stake is set on the 
center line where it passes from cut to fill, also on each 
of the base lines if the contour crosses the line obliquely, 
and a section is required at each to define the terminal 
pyramids. The grade stakes are located by trial on the 
ground, using the grade-rod reading. Fig. 119 illustrates 
this case, showing the roadbed in plan and the place of 





Fia. 120. 


Fie. 119. , 


cross-sections by dotted lines. The notes of these sections 
appear in the form of fieldbook. 

When an embankment ends at a trestle the material takes 
its natural slope at the ends as well as at the sides. Fig. 
120. ABDE is the last cross-section, but end stakes are to 
be set at H and K to define the end sections taken parallel 
to the center line. The amount of material in the terminal 
wedge and quarter-cones may then be computed and added 
to the rest of the earthwork. 

258. Other Methods. Where the transverse slope of 
the ground is steep it may be impossible to locate the slope 
stakes at one setting of the level; sometimes several settings 


CROSS-SECTIONS 225 


may be required, involving finding a new H.J. and a new 
grade rod each time. In such cases the section may well 
be taken with a level board and rod, beginning ‘with d at 
the center line and stepping up or down from the center line 
as far as the section may require. Here, as before, the true 
place of the slope stake is where the distance from the center 
and the depth to grade satisfy the formula, eq. (332). 

When the transverse slope is quite regular within the limits 
of the section its slope ratio, s’, may be ascertained and used 
to find approximately the distance to the required slope 
stake. This done in advance will save several trial com- 
putations on the way in case of a deep section, by whatever 
method the section may be staked out. For this purpose, 
_ find the depth, BF, at the edge of the base; then in Fig. 
116, 

BH = sh = s'(h — BF) 
from which 


h= BF 





s’ —s8s 
and this in eq. (332) gives 





C= (334) 
which applies to the deep ebb in _ cuts or fills. 
Similarly, for the shallow side, 
de 5) (335) 





259. Where a number of consecutive sections have been 
taken by level board it is well to run a special line of levels 
along the slope stakes, verifying or correcting h and conse- 
quently x in each section, and resetting a slope stake when 
necessary, at the same time correcting the section notes. 
All benches have been verified by the test level survey, 
but occasionally a surface elevation may be found in error 
on the center line, especially at a plus between stations 
where a cross-section is desired. Level board work should 
start with a correct value of d. 

260. Form of Field-book. A complete record of 
all cross-section work is kept in the cross-section book. 
On the left-hand page is recorded, in the first column, the 
numbers of the stations and other points where sections 
are taken; in the second, the elevations of those points, 


226 FIELD ENGINEERING 


copied in part from the location level-book, but verified or 
corrected at the time the section is taken; in the third, the 
elevation of the grade for the same points; in the fourth, 
the width of base b; in the fifth, the slope ratios, s; and in 
the sixth, the grade rod. The right-hand page has a central 
column, in which, and opposite the number of the station, 
is recorded the center depth of the section, marked + or —, 
to indicate cut or fill, as the case may require. To the right 
of this are recorded the notes of that portion of the section 
which lies on the right of the center line, as the line was run, 
and to the left, the notes of the left side. The distance 
from the center to each point noted is recorded as the nu- 
merator of a fraction, and the cut or fill at the point as the 
denominator prefixed by a + or —, as the case may require. 
The denominator for a grade point is zero. The numbers of 
the stations should increase wp the page, as in a transit book, 
so that there may be no confusion as to the right and left 
side of the line. The several points being noted in order 
as they occur from the center outwards, the notes farthest 
from the center of the page usually appertain to the slope 
stakes; but in case the cross-profile is extended beyond the 
slope stake, the note of the latter should be surrounded by 
a circle to distinguish it. The following form is a specimen 
of a right-hand page, with the first column only of the left- 
hand page: 





9.8 12.8 

125 1.2 wef aig. 
8 11.3 

ae ats as 2 

+40 0 ‘ —2.2 


Sta. L C R 


14.2 04. 


sad ~0.8 See 
11 42.8 aig a =18 
14.5 an 9.8 


+94 


16.0 7.0 ie tt 

16.0 _7.0_ 1.4 oh 

2A +4.0 $2.0 2 0 
20.5 12.0 13.6 

0.5 12.0 3.0 ge, 

‘0 47.0 44.6 i $2.4 





CHAPTER XII 


CALCULATION OF EARTHWORK 


261. The first step toward finding the cubical content 
of an excavation is to divide it into a number of prismoids 
by several cross-sections. 

A prismoid is a solid having plane parallel bases or ends, 
and bounded on the sides either by planes, or by such sur- 
faces as may be generated by a right line moving continuously 
- along the edges of the bases as directrices. 

The’ positions of the cross-sections must be so selected 
that the solid included between any two consecutive sections 
may be a prismoid as nearly as possible. Upon a tangent 
the roadbed and side-slopes are planes, so that the pris- 
moidal character of a given solid depends upon the shape of 
the natural surface. When the natural surface is a plane, 
the sections are taken only at the regular stations, 100 feet 
apart; when it is curved, warped, irregular, or broken, 
the sections must be more numerous, so that the surface 
limited by any two shall be composed substantially of right- 
lined elements extending from one section to the other. 

If two end sections of a prismoid are somewhat similar, 
we infer that the corresponding points are connected by 
right-lined elements, forming in each case the axis of a ridge 
or of a hollow. If one section has less breaks than the next, 
some of these ridges or hollows must vanish; and in order 
that the solid may be a prismoid, they must vanish in the 
section of least breaks; therefore a cross-section must be 
taken on the ground through the point where each ridge 
or hollow vanishes, and the distance of that point from the 
center line noted, so that it may be coupled with the proper 
point in the next section for exact calculation of content. 

When ridges or hollows run diagonally across the line of 
road, cross-sections must be taken where they are intersected 
not only by the center line, but also by the side slopes; that 
is, sections must be taken so that a side stake may stand 

227 


228 FIELD ENGINEERING 


on top of each ridge and at bottom of each hollow. In case 
the center line intersects at right angles a retaining wall or 
other vertical surface, two cross-sections are required at the 
same point, one at top and the other at base of wall, in order 
to furnish the data necessary to calculate the content cach 
way from the vertical surface. (See § 253.) 

Every thorough cut terminates in either side-hill cutting, 
a pyramid, or a wedge; the latter happens only when the 
contour of the natural surface is at right angles to the line 
of road. Sections should always be taken through the points 
where the edges of the roadbed meet the surface, as_ these 





HIG.) 120: 


¥ 


are the points of separation between thorough and side- 
hill work. An illustration of such a case is given in Fig. 
121. This sketch conforms approximately to the notes 
taken in § 260. In side-hill work the foregoing rules apply 
as well, but sections will generally be more numerous than 
in thorough cuts. The same rules apply also to embank- 
ment, but as grading is preferably paid for in excavation, 
the same precision in determining the quantities in embank- 
ment is not usually necessary. 
262. Formuias for Sectional Areas. 
Let 6 = base of section or width of roadbed; 
horizontal. 
vertical ’ 
d = depth at center stake; 
h, k = depths at side stakes; 
m,n = horizontal distanees from center to side stakes. 





s = slope ratio = 


CALCULATION OF EARTHWORK 229 


For ground level transversely, the section is a trapezoid 
and the area is 


A = bd + sd? (336) 
or directly from the field notes, 
; A=3(b+m+n)d (337) 


For ground of uniform transverse slope between slope stakes, 
Fig. 122, the section consists of the trapezoid ABOE and 
the triangle HOD. Hence 

A =}(AB+ EO)EN + }HO(DH — EN) 
A = 3(AB.EN + EO.DH) 











or 
A =3|bh +k(b+ 2sh)] | 
also (338) 
A = 4[bk + h(b + 2sk)] 
From which also 
A=tbh+mk ) 
and (339) 
A = 3bk + nh 
D 
| 
| 
B 
: 
h 
! i 
i Pa RRR A G B H 


Fie. 122. 


These formulas are independent of the center depth. They 
are convenient for calculating the area of a plotted section 
having an irregular surface after the surface line has been 
averaged by stretching a silk thread over it. The points 
where the thread intersects the slope lines determine the 
values of h,k, m, and n respectively. 

When the ground has uniform slopes transversely from the 
center lo the side stakes: Fig. 123. If in the diagram we 
draw HG and DG, the section will be divided into four 
triangles, two having the common base CG = d and respect- 
ive heights GN = m and GH = n, and two having the equal 
bases AG = GB = $b and the respective heights HN =h 
and DH =k. Hence we have for the area of section 


A = 3d(m + n)+2b(h + k) (340) 


230 FIELD ENGINEERING 


Otherwise, if the slope lines are produced to meet below 


grade at P, then GP = a cig The area of CEPD is 


26: 





eCb XN NHe= i(a+d oy om +n). The area of ABP is 


INCE ee a Hence we have for the area of the section 


A= i (4 ob 5 5, (m +n) — — (341) 


Both these formulas are convenient, and as the values 
of the several letters can be substituted directly’ from the 
field notes, it is unnecessary to plot such sections. 


Pee ete ee NS 





dies. IB. 


The triangle ABP is called the grade triangle. Computa- 
tion by (341) is somewhat more rapid than by (240), since 
hé and - are constants for the same base and side slope. 

Piront elevations are taken at points immediately above A 
and B, in Fig. 123, in addition to those at C, D, and E, the 
section is called five-level. The area may be found as ex- 
plained for Fig. 124. 

When the surface of the ground is irregular the plotted 
section may be divided into trapezoids by verticals drawn 
through the surface breaks and at the slope stakes. The 
area of the section is then the sum of the areas of these 
trapezoids less that of the two outside triangles. Fig. 124. 

Or the section may be further divided into triangles, one 
vertical line serving as the base of two triangles which are 


CALCULATION OF gg teuateronit 231 


computed in pairs. From the sum of all the areas it may 
or may not be necessary to deduct the area of one or both 
of the outside triangles. With either of these methods the 
dimensions used are taken directly from the field notes, and 
many times it is not necessary to plot the section. 

But when the breaks are numerous the number of parts to 
be calculated may be greatly reduced by the use of one or 
more averaging lines stretched over the surface line of the 
plotted section and noting the points where they intersect 
the center line and the side lines. Such points then govern 
the section in place of the original notes, and the section 
is thus graphically reduced to a two-level or three-level 
section, as the case may require, with the same ultimate 
_ results in area or solidity. 





The planimeter is sometimes used in determining the 
areas of irregular sections. Its use will prove economical 
in case of double-tracking work. It should never be em- 
ployed where disputes are likely to arise. 

In side-hill work, the section may also be divided into 
triangles, and the area found as with irregular sections. 

Valuation work. In work on state or federal valuation 
of existing railways, it is often necessary to estimate vol- 
umes where the original profiles and notes of the cross- 
section have been either lost or destroyed. Under such 
conditions, the side heights A and k, together with the width 
of roadbed, b, and the slope ratio s, are obtained from direct 
measurements in the field. No information as to the con- 
dition of the original surface is available, and consequently 
it is considered as sloping uniformly between slope stakes. 
When the existing roadbed is level, eqs. (338) or (339) will 
apply. If the roadbed is not level transversly, the most 
satisfaetory method is to use a level line at the top of rail or 
at the top of tie as a reference line. The area between this 
level line and the line joining the slope stakes may then be 


232 FIELD gNenerne 


found, and the resultant area obtained by adding or sub- 
tracting the proper amounts for subgrade level. 

263. The volume of a solid may be found either by the 
mean area formula or by the prismoidal formula. The former 
does not in general give the true volume, while the latter 
does. , 

The mean area formula is 

gab At) 

27 2 
where S = the volume in cubic yards between two adjacent 
sections, | = the length in feet, A and A’ = the areas at the 
two parallel ends. . 

Equation (342) may also be expressed as 








(342) 


This is based on the assumption that each area extends a 
distance of $l. 

The prismoidal formula is 

et l 

6 X 27 
where M =the area of a section midway between the ends, 
and the others are as before. This area is not a mean of the 
other two, but the linear dimensions of the mid-section 
are means of the corresponding dimensions of the end sec- 
tions; from which, therefore, the area of the mid-section 
may be computed. 

Computation of Volumes. The labor of calculating the 
prismoidal volumes .may, in many instances, be Jessened 
by first using eq. (3842) and then applying a correction, 
called the prismoidal correction, The result will be the 
same as if eq. (343) were used. This correction formula 
will now be found. 

264. Prismoidal Correction Formulas. Let C = the 
difference between results in computing the contents of a 
solid by mean areas and by the prismoidal formula. 

Then 


(A +4M 449 (343) 





CG = Sx a Sp (344) 
where 


Sz = the mean area volume, bd 
and 


Sp = the prismoidal volume, 


CALCULATION OF EARTHWORK 233 


From eqs. (842) and (343) there results 


Case 2M + A’) (345) 


which is the general formula for the prismoidal correction. 
Triangular Section. Fig. 125. When the bases of a solid 
are triangles, A = 3bk; A’ = 4b’k’; and 


w= 4| C5") ES") 


Substituting these quantities in eq. (845) and collecting 
terms there results 





l / / ‘ 2 
Os eat (346) 





‘ae 


ENG ao. 


The prismoidal volume may in this case be obtained 
directly from eq. (343) rather than by use of eqs. (342) and 
(346). By substituting in eq. (343) the values of A, A’, and 
M just found, and collecting terms, there results 


l 


Sp = 19097 


[(2b + b’)k + (2b'+ b)k’] 

Three-level Sections. A solid whose bases are three-level 
sections, may be divided into triangular prisms, and the 
mean area volume of each prism found. For the prismoidal 
corrections, eq. (3846) may be applied to each triangular 
prism. Referring to Fig. 123, the correction for the tri- 
angular prisms between AGH, BGD and similar triangles 
at the second station will be zero, since b and b’ are equal. 
The correction for the two triangular prisms bounded by 


934 FIELD ENGINEERING 


| GEC, GDC and similar triangles at the second station will 
be 


if " 
pal a Sb as _ / ( ot / 
TOOT cs ek ee 
and 
l ; ; 
Pa AE oat) 
or for the part EGDC it will be 
Cg labie con on eee 


12, % 27 
Designating (m + n) by D, and (m’ + n’) by D’ this becomes 


C = a5 557 Gd) = D’) (348) 





Fie. 126. 


Again by substituting the values of A, A’ and M from 
eq. (341) in eq. (345), the same result will be obtained. 

When C as found from eq. (348) is positive, it signifies 
that Sz is larger than Sp, and vice versa. Heeeben (344) 
may be written Sp = Sz — C. 

A general rule may be formulated as Hic 

Thé prismoidal correction in’ general is to be subtracted 
from a volume computed by the mean area method. The 
only exception is when, in the end sections, the greater center 
depth is combined with a smaller distance between slope 
stakes; a case of rare occurrence, but which changes the sign 
of one factor in eq. (348) and therefore of C. C is then to be 
added. 


CALCULATION OF EARTHWORK 230 


265. Application of the Prismoidal Correction. In 
the case of three-level sections, or with wedges, the data from 
the field notes are applied directly in eq. (348). 

Passing from Cut to Fill. Figs. 126 and 127 approxi- 
mately illustrate part of the notes given in § 260. 

The volumes between stations 10 + 68 and 10 + 94 are 
represented in Fig. 126. The solid B’CBD is a triangular 
pyramid, the volume of which is equal to one third the 
product of the area of the base and altitude, and the pris- 
moidal correction is zero. The correction for the solid 
A'B’O'E'EAC as found from eq. (347) is 

Gist aver (d — O)|(m +n) — (m’ + 0)] 

Side-hill Work. Between stations 10+ 94 and 11, Fig. 

127, there is a triangular prism on the left (omitting from 





Fre. 127. 


consideration the reading at the point 7 feet from the center) 
and a prismoid on the right. The correction for this prism 
is given by eq. (346). The prismoidal correction for the 
prismoid on the right is found from eq. (347) which gives 


l 
cha’ x 27 


The sign of C in this equation will be positive, thus showing 
that the order in which the sections are taken is immaterial. 

Irregular Sections. The volume between two irregular 
sections may be found either by the prismoidal formula 
or by the mean-area formula. In the latter case, there are 
several approximate ways of determining the prismoidal 
correction. The two in greatest favor are: 

(1) When the ground is only slightly irregular, neglect 


vu 


(0 = d')[(m +0) —(m' #.G'L') 


236 FIELD ENGINEERING 


the intermediate points, and for the correction, treat the sec- 
tions as three-level. 

(2) Plot the sections on cross-section paper and draw 
lines which will form an approximation to the three-level 
sections. Find d and D for each section and use eq. (348). 

The first method will, in general, give results sufficiently 
close. Personal preference governs largely the choice of 
’ other methods. 

266. Correction of Earthwork for Curvature. 
The preceding calculations are based on the assumption that 
the center line is straight, with cross-sections at right angles 
to it. When an excavation is on a curve, the cross-sections, 
being in radial planes, are inclined to each other, so that the 
condition of a prismoidal is not exactly fulfilled. But by 
the property of Guldinus, if any plane area is made to revolve 
about an axis in the same plane, the volume of a solid gen- 
erated by the area is equal to that of a prism having a base 
equal to the given area, and a height equal to the length of 
path described by the center of gravity of the area. The 
path, being the are of a circle, is proportional to the radius 
drawn to the center of gravity. If therefore a cross-section 
is symmetrical with respect to the center line, the path of 
the center of gravity is equal to the measured length of the 
center line, and no correction for curvature is required. 

But when the ground is inclined transversely, the center 
of gravity is one side of the center line, and its path, if we con- 
ceive it to sweep around the curve, from one end of a pris- 
moid to the other, is longer or shorter than the distance 
measured on the center line, according as the center of gravity 
is outside or inside of the center-line curve. 

Let C = correction in cubic yards due to curvature; 

S = cubic yards as obtained by prismoidal Pryoile 
R = radius of center line; 
e = eccentricity of center of gravity of section; 
horizontal distance from center line to center of 
gravity. 


I 


I 


We then have the proportion, 
Sone Guta sy Reke! wR 
Se 


C= R (349) 


CALCULATION OF EARTHWORK 237 


As the sections of a solid are seldom similar and equal, 
we shall usually have a different value of e for every section, 
from which, however, a mean average value may be deduced, 
and used in the above formula. But it will be more con- 
venient to correct the areas themselves for eccentricity 
before finding S, which will then require no correction. For 


the same result will ensue whether we multiply S by a 


- or multiply one of the component factors of S by the same 


ratio. 
If then c = correction of area in square feet to eccentric- 


ity, we have at once 


Ae 
R 


C= 





and the corrected area equals A + ¢ according as the cut is 
deeper on the outside or inside of the curve. Each area 
used in determining the solid contents should, on a curve, 
be first corrected in this manner. 

To find the value of e for any three-level section. Fig. 128. 

Find the areas either side of the center line separately, 
calling them H and K, and take their sum and difference. 
Using the same notation as in § 262, H = 3md + jbh, 
K = ind + ibk, andH+K =A. f 

K:.— H = 3d(n — m) + ib(k — h) 

In the figure draw CE’ equal to CE, and the triangle CE’D 
will represent the area (K — H). Bisect the side H’D, and 


draw a line from C to the middle point. Then the center of 
gravity of the triangle will be on this line at two thirds its 


238 FIELD ENGINEERING 


length rrom C, and the horizontal distance of the center of 
gravity from C is } X 3(m +n) =3(m-+n). The center 
of gravity of the remainder of the section is on the center 
line CG, so that the value of e is found from the proportion 


Pee caer) wee ard mime | 


Dt an 
e= 34 (K — A) 





Ae _n+m,, : 
Hence ¢ = = “35° [id(n — m)+ 4b(k—h)] (350) 





Sections which are more irregular may be plotted and 
reduced by averaging lines to three-level sections, in order 


v 





te ee ee ee ae ee 


Fig. 129. 


that the formula may be applied. If the ground is so irregu- 
lar as to require the computation of the middle section, 
the correction c should be found and applied to this area 
(M) also before introducing it into the prismoidal formula. 
As the correction for curvature is always relatively small, 
it is usually ignored in practice for thorough cuts, except 
where deep cuttings with steep transverse slope occur on 
sharp curves. , 

The correction is of more importance relatively in 
side-hill work as the center of gravity of the section is 
more remote from the center line. Let the section be reduced 
to a triangle by an averaging line (Fig. 129), and w be the 
base of the triangle formed by the averaging line. The 
center of gravity is at one third the horizontal distance from 
the middle point of w to the side stake D, while the distance 


CALCULATION OF EARTHWORK 239 


of this middle point from the center stake C is evidently 


BU) Ze 
Hence ES 
e = 7b — zw + alin — (3b — 5w)] 
or 
e=3(b +n — w) 
and 


Be A Cie Ur te ee Wie 
Ci R = yee xa (351) 
The correction c will be plus or minus as before explained. 
This formula applies to all side-hill triangular sections, 
whether there be cut or fill at the center stake. 
Example 1.—Thorough cut; base 20; slopes 1}: 1. 


1 = 100; 8° curve, left; R = 716.78 


16 58 
if ee enone 
Notes. Ani + 12 +33 

ae 40 

Sok WOR Re aT 
Then K = 4 X58 X12 +14 X 20 X 32 = 508 

H=1xX16X12+1x20x 4=116-.A = 624 
K-—H= 392 
_ °16 + 58 x 

Hid: (849) 08. Sarma gig 892 = 13.49 





(A+c) 637.49 
K'’=4X 40X84 X 20 x 20 = 260 
Mitte Is OG 5 ft x 20rMN as O02) A oo! 


198 
198 = 4.87 


(A’ +c’) = 326.87 
From which we obtain S = 1758 cu. yds.—Ans. 


a 
| 
a 
| 


i340 
Sap 716.78 


c! 


Without correction we have 1726 cu. yds. 


Showing a difference of 32 cu. yds. 


Had the curve been to the right with same notes, ¢ would 
have been minus, and S would = 1694. 


240 FIELD ENGINEERING 


Example 2.—Side-hill cut; base 20; slopes 13: 1. 
L = 60; 10° curve, right; R = 573.69 _ 








6 40 
Notes 0 + 2.8 + 20 
2 37 
ee semaines: 9 
A =X 16x20 = 160 
Lirias “ene L; 
Kq. (350) © = 3 573.69 160 = 3.08 
(A —c) = 156.42 
A’ =; X84+.18 = 72 


Pe 20 ieee 


= 3x 57309 (27> ioe) 








(A’ —c) = 69.95 
Hence S = 248 cu. yds. 
Without correction S would = 255 cu. yds. 


Difference 7 cu. yds. 


* 


267. Any isolated mass of rock or earth which oc- 
curs within the limits of the slope stakes, but not included in 
the regular notes, is separately measured and noted, so that 
its contents may be computed and added to the sum of the 
same material found in the cross-sections. 

268. Borrow-pits.—When the excavations will not 
suffice to con-plete the embankments, material may be taken 
from other localities, termed borrow-pits. These should be 
staked: out by the engineer and their contents calculated, 
unless the contractor is to be paid for work by embankment 
measurements. A number of cross profiles are taken of 
the original surface, and (on the same lines) of the bottom 
of the pit after it. is excavated, which furnish the depth of 
cutting at each required point. Borrow-pits should be regu- 
larly excavated so that they may not present an unsightly 
appearance when abandoned. Borrow-pits may be avoided 
by widening the cut uniformly at the time it is staked 
out, so that it may furnish sufficient material; provided 
the material is suitable, the embankment accessible, and the 
distance not too great. When the excavation is in excess, 


CALCULATION OF EARTHWORK 241 


the surplus material should be uniformly distributed by 
widening the adjacent embankments, if possible; otherwise 
it is deposited at convenient places indicated by the engineer 
and is said to be wasted. 

269. Vertical Prisms. Material excavated from bor- 
row pits may be measured also by first staking off the field 
into rectangles of suitable size and taking the elevation of 
surface at’every stake. If convenient some triangles may 
be laid out where boundaries are oblique. All lines should 
be carefully referenced. After excavation the intersections 





Ine, UO), 


are relocated and their elevations taken; the difference in 
elevation being the depth of cut at each point. Thus the 
entire mass is subdivided into vertical prisms, truncated or 
otherwise, the cubic contents of which may be easily com- 
puted. To facilitate the reduction to. cubie yards it is 
customary to choose such dimensions as will make the’ area 
of each prism base some multiple of 27 square feet. 

The proper size of rectangle to be used depends upon the 
natural surface which should give practically right lines to 
prism ends. Shallow work calls for larger rectangles because 
so soon excavated. 

270. To find the volume of a truncated triangular prism, 
whose horizontal section is a triangle of given area A. Fig. 


130. 
Let hi, ho, and h3, denote the heights AD, BG and CH. 


DAS FIELD ENGINEERING 


Through the lowest point D of the upper face, draw a 
horizontal plane DEF, cutting off the pyramid DEGHF, 
whose base is a trapezoid, and whose altitude is the per- 
pendicular distance from D to FE and designated by a. 

The volume of the pyramid DEGHF 

3a X FEGH = 3a X3(EG+ FH)FE = ZA(EG + FA), 
since 
34 X FE =A. 

By geometry, the volume of a prism, not truncated 
= Ah = Ah in this case. The volume of the truncated 
prism ABCDGH = the sum of the two parts 


= Ah, + 3A(EG + FH) 
== AUX all AtieEG):-- Ga FH) 
or, in cubic yards, 
S _Ah+the+hs 


== : (352) 





Imelg USBES 


271. To find the volume of a truncated rectangular 
prism, whose horizontal section is a rectangle of area A. Fig. 
131, 

The end area formula gives the correct cubic content, 
since in this figure the prismoidal correction is zero, provided 
that, the top and bottom ends, whether plane or warped, 
are bounded by right lines. 

Let the heights AH, BF, CG, and DH be represented by 
hi, he, hs, and hg, and the area ABCD by A. 


CALCULATION OF EARTHWORK 243 


Now the figures ADHE and BCGF may be considered as 
bases of the solid with the altitude as AB = DC. 
The mean area volume 


AB (area ADHE + area BCGF) 
22 


AB | (5) ap + (2E*) ze| 


re B (hi +h + ha + ha) BC 


S = 








pole 





But 
AB X BC= re 


-2(" + he as hs aii ) (353) 


























BiGao Loc 


272. Method by Unit Areas. When anumber of prisms 
contiguous to each other have the same horizontal section, 
A, their total cubic content may be obtained by a single 
formula. For since by eq. (353) the four-corner depths -of 
a prism are to be added together, it 1s evident that the depth 
at any corner whatever will be taken into account as many 
times as the number of rectangles which meet at that point. 

Thus in Fig. 182 the depths at the corners a, bi, C2, ds, 
etc., enter into the volume of one prism only; while those 


244 FIELD ENGINEERING 


at dz, a3, a4, ete., affect two prisms each; and those at 
be, 3, G4, ete., affect three prisms each, and so on. : 

Therefore if we let s; = the sum of all depths used but once, 
s, = the sum of all depths used twice, s; = the sum of all 
derths used.three times, and s, = the sum of all depths 
used four times, we shall have for the total content of all: 
prisms considered, 


re S: + 2s. + 38; + 484 
total = As 4X 27 : 


Marginal masses, not included in this system, must be 
measured and computed separately. 

273. The foregoing method of prisms is not well adapted 
to steep hillsides or bluff banks from which material is to be 
borrowed. In such cases if the pit is apart from the road, 
a base line is staked out in the general direction of the con- 
tour, and frequent cross-sections are taken to define the shape 
of the pit before and after excavation. 

If the needed material is to be taken from the regular 
cuts the regular cross-sections are extended beyond the slope 
stakes and other stakes are set to mark the limit of the pit, 
and the quantities are computed in the usual manner. 

But if the roadbed consists of a long shallow embank- 
me2zt made of material scraped up at random from either 
side, the quantities should be measured arid paid for in the 
finished fill, under the terms of the contract, without refer- 
ence to the borrow-pits. 





(354) 


CHAPTER XIII 
EARTHWORK TABLES 


274. The labor involved in the calculation of earthwork 
- may be greatly reduced by the use of earthwork tables. 

The common forms of tables are those for level sections; 
three-level sections; triangular prisms; and_ prismoidal 
corrections. 

275. Level-sections. For purposes of preliminary es- 
‘ timates, tables computed on the assumption that the ground 
is level transversely are quite sufficient. The volume of 
any one solid may be greatly in error, but it is surprising 
how nearly the total of the preliminary volumes on an ex- 
tended line, will compare with the final volumes obtained 
after cross-sectioning. The center cut or fill is taken from 
the profile. 

The area of a level-section is expressed by eq. (836) which 
is 

A = bd + sd? 

~ The volume in cubic yards for a length of 100 feet is 


S= aul (bd + sd?) (355) 

Table XXX was computed by this formula with such 
values of 6 and s as are in common use. 

A section affects the volumes for 50 feet each side of 
itself. For preliminary estimates, the volumes may remain 
as taken from the table in which case, where there are whole 
stations, the volume extends from plus 50 to plus 50. In 
the construction of the mass diagram (see Chapter XV) 
the volume for a 50-foot length should be added to that for a 
preceding 50-foot length, and again to a succeeding 50-foot 
length, in order to give the amounts between adjacent 
whole stations. For substations the proper percentage is 
taken. 

245 


246 FIELD ENGINEERING 


276. Tables of Triangular Prisms. The volume of 
a triangular prism is equal to the product of the area of the 
base and the altitude or 


bal 
i DESEO, 





in cubic yards. (356) 
where b and a are the base and altitude of the triangular 
section, and / is the altitude of the prism. 

For a 50-foot length this equation becomes 


f . 


Table XX XI has been computed from eq. (357) for dif- 
ferent values of 6 and a. In this table b is considered as 
“width” and a as “ height,” but these terms may be used 
interchangeably when convenient, since the product is the 
same. 

The volume of any solid which can be divided into triangular 
prisms may be found from this table. 

277. Tabies of Three-level Sections. The area of a 
three-level section ntay be found by the two methods given 
in § 262. In the first method the section is divided into 
four triangles. The volume in cubic yards for prisms of 
length 50 feet will be 
50, 
5d? 

It is to be noted that the terms on the right-hand side of 
this equation are similar to eq. (857) for the volume of a 
triangular prism. Consequently Table XXXI may be 


Sr. = 27d (m-+n)-+ b(h+k) (358) 


50 
used for the products of the parts and the constant Bd 


In the second method of § 262, the area is made up of two 
triangles minus the grade triangle. The volume for length 
50 will be | 
50 b? 
| Bd 2s 

Table XXXI of Triangular Prisms is also used for this 
method. 

Extension of Level-section Table. Where the center cuts 
and fills are taken from the profile for a preliminary esti- 


Sx = 2 (1 it | on iit (359) 


EARTHWORK TABLES 247 


mate and the quantities obtained from Table XXX, these 
may be corrected by use of Table XX XI after the cross-sec- 
tions have been taken, which give the side heights. These 
corrections are illustrated by the triangular prisms with bases 
DCD’ and CEE’ in Fig. 133. The base of each triangle 
is equal to 4b + sd, and the altitudes of the triangles are 





Gama 


the differences between the center height and each side height. 
If a side height is less than the center height, the correction 
is to be subtracted and if greater, added. The products 
of the bases and altitudes of the triangles and the constant 


a give the correct volumes on each side. 


It should be noted that Table XX XI is for 50-foot lengths, 
while Table X XX is for 100-foot lengths. 

278. Tables for Irregular Sections. A _ table of 
triangular prisms is suitable for finding the volumes of 
earthwork solids with irregular sections for bases. The 
sections may be divided into triangles as in Fig. 134, and 






| 
| 
| 
cn 
pe 


| 
| 
! 
» 
) 
1 
| 
Me CS 
J 


Fie. 134. 


the bases and altitudes of each triangle found. Equation 
(357) applies to each partial volume, and the total volume 
is the sum of the parts. Table XX XI is available for this 
work. 

279. Prismoidal Correction Table. Where the mean 
area formula is used for finding the volume of a solid, as is 
the case with the tables described above, the prismoidal 


248 FIELD ENGINEERING 


correction must be applied to give the true volume. Eq. 
(848) gives the value of this correction. For a distance 
between sections of 100 feet and for the result in cubic. 
yards, this equation becomes 


1 / 
C = goqd — @)(D — D’) (360) 
Table XXXII has been computed from this equation for 
different values of (d — d’) and (D — D’). ‘The correction 
is applied as explained in §§ 264 and 265. 
280. Example. Use of Tables XX XI and XXXII. 





91.0 19.2 
Notes: Sta. 3 +380 arate +68 
Crt 15 0 

1 > —$ $$ ft 


The plus sign merely indicates cut and is not considered 
in the computations. 

First, by the method based on eq. (358). 

For station 2,-d = 45; m+n = 32.1; b= 9.0; and 
h+k=9A. 

From Table XX XI, 





Opposite H’g’t. 4.5 under 3 take 10 X 12.500 = 125.000 

i ila? hac 1X" 81333 =" 87339 

ne eli seapton Case Ose 417 

Opposite H’g’t9.4—$£--" 9, “1a X78..338 = 18383 

Total volume for length of 50 feet = 212.083 
For station 3, d = 7.38; m+n =40.2;,b =9.0; and 


h+k = 148. 
From Table XX XI. 
Opposite H’g’t. 7.38 under 4 take 10 X 27.037 = 270.370 
SE ae OG leap LO ee aoe 
Opposite H’g’t. 14.8° “ 9 ** 1X 123.333 =123 333 


Total volume for length of 50 feet = 395.056 


The quantities may be obtained from the tables some- 
what more easily by not strictly adhering to the terms 
“height ’’? and ‘ width,” 


EARTHWORK TABLES 249 


Consequently the mean area volume between stations 
2 and 3 is 
Sz = 212.083 + 395.056 = 607.139 
The value of C may be found from Table XXXII, which is 
based on eq. (360). 
Now d —d’ = 2.8, and D — D’ = 7.1. 
and 


1 
394 KZ SUXK0G, a6. b3d 


Sp = Sz — C = 607.139 — 6.185 = 601.004 


Second, by the method based on eq. (359). 
b 

Now ve 6.0. 

b 


For station 2, d + pee 10.5 and m + n = 32.1 


From Table XX XI, 
Opposite H’g’t. 10-5 under 3 take 10 X 29.167 = 291.670 





‘2 1% 19.444 = 19.444 
BOF ly tere ON Deas OG 
Total = 312.086 
: : 50 
For the grade triangle there is subtracted Bd x6X 18 = 


100, giving the resultant volume for section at station 2 
as 212.086. 


For station 3, d + = = 13.3, and m +n = 40.2 


From Table XXXL 
Opposite H’g’t. 13.3-under 4 take 10 X 49.259 = 492.590 
BEND TCE 247630 2.463 


Total 495.053 


The grade triangle portion (as above) = 100. 

Then the volume of the prism with base equal to the sec- 
tion at station 3, is 395.053 

.. the mean area volume of the solid between stations 
2 and 3 is 607.139. | 

The prismoidal correction is found in the same way as in 
the first method and is 6.135, 


II 





250 FIELD ENGINEERING 


the prismoidal volume = 607.139 — 6.135 = 601.004. 

This value is the same as that found in the first method. 

The tables are carried to three decimals, thus making them 

more accurate when multiplied by 10. Work may be carried 

out to any desired degree, decimals being omitted in the final 
quantities. 


Remark.—The second method is somewhat shorter than 


the first, but where aa is not in exact decimal, the results 


fd 


obtained by the two will not exactly check. 


CHAPTER XIV 
EARTHWORK DIAGRAMS 


281. General Principles. An equation of the first 
degree containing but two variables can be graphically 
represented by a straight line. Equations of the second or 
higher degree can only be represented by curves. 

Consider the equation 

(t piceaua tp (361) 
Where a is a constant and x and y the variables. This 
equation is that of a straight line and is represented by Fig. 








HIG: elas Fia. 136. 


135 for a certain value of a. Various values of zx are laid 
off on the axis of « and the corresponding values of y are 
computed and laid off at right angles to OX. The line 
joining successive points will be represented by the line 
OA, which passes through the origin. Many problems can 
be solved from diagrams based on eq. (861), among which 
may be mentioned that for computing acres of right of 
way. 
Another equation is of the form 


Y= 2x (362) 
251 


252 FIELD ENGINEERING 


Here there are three variables, but if we consider one as a 
constant, we may represent the equation graphically by sub- 
stituting different values for the second variable, and solve 
the equation for the third quantity. Fig. 136 represents 
solutions of this equation for z of 3, 1 and 2. When z = 3 
and x = 1, y will equal 3; with z = } and z = 2, y will 
equal 1, and so on. The z = 3 line is thus found by laying 
off the quantities x and y along and at right angles to the 
axis of X. Cross-section paper may be used to advantage 
in work of this nature. 
Other common equations are of the form 


y = aer (363) 
y =a(z+ b)(@ + c)k (364) 


where a, b, c and k are constants. Diagrams for equations 
in form similar to both of these may be plotted as shown 
in Fig. 136. 
Nearly all the eauiations for the computation of earth- 
work are similar in form to the last three equations above. 
282. Triangular Prisms. The volume in cubic yards 


of a triangular prism of length 50 feet is ba, where 6 and a 


represent the base and altitude of the triangular section. 
A diagram may be constructed for solving this equation. 
Different values of b and a may be substituted, giving corre- 
sponding volumes. 

The table which follows will be of assistance in plotting 
the diagram. The particular values chosen for b were taken 
for simplicity in computing and plotting. 

TABLE FOR PLOTTING PLATE I.—TRIANGULAR PRISMS 
VALUES OF b 

















0 5.4 10.8 16.2 21.6 27.0 
0 0 0 0 0 | 0 0 
1 0 5 10 15 | 20 25 
2 0 10 20 30 40 50 
8 3 0 15 30 45 60 75 
‘6 4 0 20 40 60 80 100 
m 5 0 25 50 75 100 125 
zB 6 0 30 60 90 120 150 
3 vi 0 35 70 105 140 175 
- 8 0 40 80 120 160 200 
9 0 45 90 135 180 225 
10 0 50 100 150 200 250 








EARTHWORK DIAGRAMS 253 


Cross-section paper may well be used for the diagram, but 
since the paper is lable to be more or less inaccurate as 
respects the ruling of the lines, several points in each line 
should be plotted considering the ruling of the paper as 
correct. The successive points are then connected. 

A small reproduction of folding Plate I at the end of this 
volume is given in Fig. 137. Its construction is self-evident. 






































Volume of Triangular Prism in Gu. Yds. 
Volume of Triangular Prism in Cu. Yds. 

















Values of b 
Hires 7, 


283. Use of the Diagram. Let the altitude and base 
of a triangle be 9.0 and 12.0 respectively. To find the 
volume of a triangular prism of length 50 feet. Go out 
to the b = 12.0 line and up to the a = 9.0 line. Read the 
volume as 100.0 cubic yards. This quantity, as previously 


. e O 
noted, is the product of 9.0, 12.0, and 7 


284. Diagram for Three-level Sections. The mean 
area volume of a solid of length 50 feet and with triangular 
base is found from eq. (359) which with D substituted for 
(m + n) becomes ‘ 

* 50 b 50 b? 
Since b and s are constants in any particular case, there are 


254. FIELD ENGINEERING 


only three variables in this equation, d, D and S. As pre- 
viously explained, one of these values may be considered 
constant and d will be so taken. Having chosen the con- 
stant d, values of S may be found for different values of D. 
Here, as.with triangular prisms, the work of plotting may be 
expedited by previously computing and arranging a table 
similar to the one below. 

Plates II and III are plotted for bases of 20 and,16 feet, 
and side slopes of 13 to 1. A separate diagram is necessary 
for each different combination of b and s. Values of D are 
laid off on the x axis, and those for S on the y axis. Inclined 
lines passing through successive points will be the lines 
representing the different values of d. 


TABLE FOR PLOTTING PLATE II 


DISTANCE BETWEEN SLOPE STAKES=D 




















Center . ey 
Heights d 20.0 21.6 27.0 32.4 37.8 AZ52 

5 Const. 

18.518 20.0 25.0 30.0 35.0 | 40.0 Diff. 

0 0 9.876} 48.209) 76.542} 109.875) 143.208) 33.333 

1 18.518 29.876} 68.209} 106.542} 144.875 188.208) 38.333 

2 37.036 49.876} 93.209] 186.542 | 179.875) 223.208) 48.333 

3 55.554 69.876) 118.209) 166.542) 214.875 | 263.208) 48.333 

4 74.072 89.876) 143.209) 196.542} 249.875 | 303.208) 53.333 

5 92.590 | 109.876) 168.209} 226 542| 284.875| 343.208) 58.333 

6 111.108 | 129.876) 193.209] 256.542! 319.875 | 383.208) 63.333 

tf 129.626 | 149.876) 218.209] 286 542) 354.875! 423.208) 68.333 

8 148.144 | 169.876) 243.209] 316.542| 389.875) 463.208) 73.333 

9 166.662 | 189.876) 268.209) 346.542} 424.875 503.208) 78.333 

10 185.180 | 209.876) 293.209) 376.542} 459.875) 543.208) 83.333 














The labor involved in preparing a table for this’ work may 
be lessened by observing that certain parts of the above 
equation remain constant for successive changes in the 
variable quantities. Proper substitutions may be made in 
the general formula and the constant difference in solid- 
ities found. For instance, when d = 1 and D = 20, 


50 20 50 b? 
= — 2 —_ ——_—_- 
S a (+5) Poe 
Again when d = 2, and D = 20 
» 280 (4 4.20) 09 = 0? : 
» =F, (2+ 3) 20 54 25 


EARTHWORK DIAGRAMS 255 


Subtracting S from S’ there results 


0 


; 5 
S SS tet 


xX 20 = 18.518 
The same difference will be found between d = 2, and 
di =i3. 

Then if the next value of D = 21.6 is used with d = 
and d = 2, the constant difference of 20 would be found. 

It is advisable to solve the equation for all values of D 
when d =0. The results are entered on the horizontal 
line of d= 0. After these quantities are found the numbers 
on the other horizontal lines tnay be obtained by use of the 
various constant differences. _ ‘ 

A check crosswise of the table may also be obtained by 
solving the equation for S with different values of D. For 
instance, when d = 0 and D = 21.6 


2 
Ss ar (0 +5 F) 21.6 oo. 
oO Ss 

with d = 0, and D = 27.0. 
6,00 50 b2 
Ss a (0 +5 3) 270 - 2S 

Then 
Pijig = 50 2) (2 te bs 
Ss s=a(y (27.0 — 21.6) =33.333 


Similarly for the differences in volumes when d = 1 with 
D = 21.6 and 27.0, 

PAN fe 

or 520 (2 


_9 I 
m4 ) e70 21.6) = 38.333 


A final check on the table should be made by substituting 
the largest values of d and D in the formula. In this case 
there results 

50 50 20? 


a a a 9 sree 
S a (10 +2 =) 43. 54 3 543.208 


This operation will check all sidanes in the table. 

Curve of Level Seetion—On both plates II and III a curve 
of level section has been plotted. From eq. (331) we may 
find D for different values of d, Thus for Plate II when 


256 FIELD ENGINEERING 


d=0 D = 20.0 
d = 0.4 D = 21.2 
d=1.4 D = 24.2 
d = 2.4 D = 27.2 
d=6.4 D = 39,2 


The curve passing through the points having these co- 
ordinates is called the ‘‘ curve of level section.’’? The vol- 
umes for Sec estimates may be obtained from this 
curve. 

285. Use of the Diagram. The lines and quantities 
on Plate II afe fully designated; an example will explain 
the proper method of procedure in determining volumes. 

Example.—Cross-section notes are as follows: 








16.0 11.8 

2 ca 2. 
19.0 136 

Sta. 2 auido. 
Sta. 27 +6.0 EHO 4 


For station 28, d = 2.4 and D = 2738. 


From Plate II, at the intersection of the lines d = 2.4 
and D = 27.8 we obtain a volume of 110.1 cubic yards. 

For station 27, d= :3.0'and._D \=-32.6; 

At the intersection of the lines d = 3.0 and D = 32.6 
we find a volume of 168.3 cubic yards. 

Since the diagram gives results for 50-foot lengths, the 
total volume between the two stations is 110.1 + 168.3 = 
278.4. 

The result obtained for these same notes by tables is 
278.26 cubic yards. 

The prismoidal correction may be applied to this mean 
area volume. The amount of the correction is found similarly 
to the way explained in the next section. 

286. Diagram for Prismoidal Corrections. The 
formula for the prismoidal correction has been previously 
deduced and is 

C= had — d')\(D — ” (360) 
The results are in cubic yards for a distance of 100 feet 
between sections, 


EARTHWORK DIAGRAMS QF 


Numerical substitutions for (d — d’) and (D — D’) may 
be made and the corresponding results found. It will be 
well to take (d — d’) as 0, 1, 2, 3, 4, 5, etc., and to take 
values of (D — D’) so that the corresponding C will-be a 


whole number. For instance, when (d — a’) = 1, the 
; Ly 3 
equation »becomes 3 94\P — D’), and convenient values of 


(D — D’) will be 0, 3.24, 6.48 and so on, giving C as 0, 1, 
2, 3, 4, ete. ‘ 

Similarly with (d — d’), taken as 2, 3, 4, ete. Likewise 
the equations of other (d — da’) lnes may be found; con- 
venient substitutions of (D — D’) made, and a table arranged 
for plotting the diagram, 


TABLE FOR PLOTTING PLATE IV 








d—d’ | 0 | 3.24 | 6.48 | 9.72 | 12.96) 16.20; 19.44) 22.68) 25.92) D —D’ 
0 0 0 0 0 0 0 0 0 0 
OD i ie es @ 3 cee 6 it 8 
2 0 2 4 6 8 10 12 14 16 
3 0 Bae 4 9 12 15 18 21 24 
4 0 d 8 12 16 20 24 28 32 
5 Og BOs 6 EO 15 20 25 30 35 40 
6 0 6 12 18 24 30 36 42 48 
7 ig a ae Ss | 21 28 35 42 49 56 
8 0 8 16 24 32 40 48 56 64 
9 0 u 18 27 36 45 54 63 72 

10 0 10 20 30 40 50 60 70 80 



































Plate IV at the Wack of this volume is the diagram for 
“Prismoidal Corrections.’ Lines intermediate between 
whole numbers of (d — da’) are plotted proportionally. 

For finding prismoidal corrections for irregular sections 
one of the methods described in § 265 may be employed, 
the diagram being used after the preliminary quantities 
have been decided upon. 

287. Use of the Diagram. Consider as an example 
that d — d’ = 4.0 and D — D’ = 11.0. 

Going out for D— D’ = 11.0 and following up to the 
line of d — d’ = 4.0 we read the correction in cubic yards 
as 13.6. The value of C from computation to two decimal 
places is 13.58. When either D — D’ or d —d’ are too 
large for the limits of the diagram, they may be divided 
‘by some even quantity, as 2, 3, and so on, and a correspond- 


258 FIELD ENGINEERING 


ing multiplication made after the correction has been obtained 
from the diagram. 

288. Conclusion. The four plates placed at the end 
of this volume are necessarily reproduced at a small scale. 
On construction work a much larger scale should be adopted. 
Diagrams carefully made according to the principles here 
given, will not only give a high degree of precision, but their 
use will be conducive to speed in securing results. 

The voltime for any irregular section may be obtained 
by use of Plate I, provided the section is divided into triangles 
or trapezoids, as explained in the previous chapter, 


CHAPTER XV 
HAUL AND THE MASS DIAGRAM 


289. Haul. The cost of removing excavated material, 
when the distance does not exceed a certain specified limit, 
is included in the price per cubic yard of the material as 
measured in the cutting. But when the material must 
be carried beyond this limit, the extra distance is paid for 
at a stipulated price per cubic yard, per 100 feet. The 
extra distance is known by the name of haul, and is to be 
computed by the engineer with respect to so much of the 
material is as affected by it. 

The contractor is entitled to the benefit of all short hauls 
(less than the specified limit), and material so moved should 
not be averaged against that which is carried beyond the 
limit. Therefore, in all cuts, the material of which is all 
deposited within the limiting distance, no calculation of 
haul is to be made. ; 

On the other hand, the company is entitled, in cases of 
long haul, to free transportation for that portion of the 
cutting, no one yard of which is carried beyond the specified 
limit. Therefore, this portion is first to be determined in 
respect to its extent; and the number of cubic yards con- 
tained in it is to be deducted from the total content of the 
cutting, before estimating the haul upon the remainder. 
Find on the profile of the line two points, one in excavation, 
and the other in embankment, such, that while the distance 
between them equals the specified limit, the included quanti- 
ties of excavation and embankment shall just balance. 
These points are easily found by trial, with the aid of the 
cross-sections and calculated quantities, and become the 
starting points from which the haul of the remainder of the 
material is to be estimated. 

Fig. 138 represents a cut and fill in profile. The distance 
AB is the limit of free haul. The materials taken from AO 

259 


260k FIELD ENGINEERING 


just make the fill OB and without charge for haul; but the 
haul of every cubic yard taken from AC, and carried to the 
fill BD, is subject to charge for the distance it is carried, 
less AB. It would be impossible to find the distance that 
each separate yard is carried, but we know from mechanics 
that the average distance for the entire number of yards is 
the distance between the centers of gravity of the cut AC, 
and of the fill BD which is made from it. If, therefore, 
X and Y represent the centers of gravity, the actual average 
haul is the sum of the distances (AX + BY), and this 
(expressed in stations) multiplied by the number of cubic 
yards in the cut AC, gives the product to which the price 
for haul applies. 

But the product of AX by the number of cubic yards in 
AC is equal to the sum of the products obtained by multi- 








Fig, 138, 


plying the contents of each prismoid in AC by the distance 
of its own center of gravity from A. The distance of the 
center of gravity of a prismoid from its mid-section is ex- 
pressed by the formula 


Se ol ©: ee: : 
aS 285) 
If we replace S by its approximate value, a which 
will produce no important error in this case, we have 
L A— A’ 
a) Anes ae 


n which A should always represent the more remote end 
area from the starting point A, Fig. 188. Hence, x may be 
+ or —, and it must be applied, with its proper sign, to the 
distance of the mid-section from the starting point A, before 
multiplying by the contents S. Each parug! abies is 
thus obtained. 


HAUL AND THE MASS DIAGRAM 2o1 


If instead of the areas, A and A’, the solidities, S and S’, 
have been taken out for uniform lengths, we may substi- 
tute S and S’ for A and A’ in eq. (866) to find the value 
of x. 

By a similar process with respect to the prismoids compos- 
ing the mass BD, and using the point B as the starting 
point, we obtain finally a sum of the products representing 
this portion of the haul. 

If a cut is divided, and parts are carried in opposite direc- 
tions, the calculation of each part. terminates at the dividing 
line. If a portion of the material in AC is wasted, it must 
be deducted, and the haul calculated only on the remainder. 

The specified limit is sometimes made as low as 100 feet, 
‘ sometimes as high as 1000 feet. A limit of about 300 feet, 
however, is usually most convenient, as it includes the wheel- 
barrow work, and a large part of the carting, while it pro- 
tects the contractor on such long hauls as may occur. 

290. Quantity Profile. The ordinary railroad profile 
gives to scale.all center line depths and the relative positions 
of cuts and fills, but does not show the quantities of material 
in them. The quantity profile is so drawn that its several 
areas above or below the grade line are proportional to the 
number of cubic yards in the several cuts and fills. To 
obtain this result we lay off from the grade line at each 
point where we have a cross-section a vertical ordinate, 
plus or minus, representing to scale the number of. cubic 
yards in a length of one foot due to that section. Connecting 
the extremities of the ordinates by a smooth curve we have 
the quantity profile. Upon this the center of gravity of 
each cut or fill is not difficult to find graphically; and equal 
areas, whether in cut or fill, indicate equal solidities. The 
quantity profile assists in a study of the proper disposition 
to be made of material in the construction of the road, and 
in computing the distances that some materials are to be 
hauled. 


The Mass Diagram 


291. The Mass Diagram shows by a single vertical 
ordinate the aggregate number of cubic yards involved up 
to that point, and its several areas indicate the product of 
cubic yards removed by the distance hauled. It is con- 


262 | FIELD ENGINEERING 


structed upon a horizontal zero line laid off into stations 
with the same scale as the profile. Fig. 139. 

Beginning at any assumed point of the road, and having 
computed the quantity of earthwork required between that 
point and the next, and the next, etc., we find the algebraic 
sum of these quantities by continuous addition, and lay off 
the sums so found as vertical ordinates, plus or minus, each 
at its proper point on the zero line. Joiming these ordi- 
nates by a smooth flowing line we complete the curve of the 
mass diagram. The ordinates should be tabulated in a 
systematic way, as here shown, the quantities of cut and 
fill being distinguished by separate columns as well as 
algebraic signs, providing also for side-hill sections. The 
shrinkage column contains the cubic yards of fill in the 
preceding column, increased by the estimated percentage 
required to make good the shrinkage. The last column 
contains the ordinates for the diagram. 














Cubic Yards. 

Stati Center WS ee aE | Shrmkacen. Ouse 
ation. d YJ Sums. 
Cut. Fill. 

0 = (aio 

+30 +5.0 iigiegs ce Site eee Wee NAP ofan SS oe +155 

+90 3.3 ESS te lf Sn a tly RO al +354 
] +2.6 OAee ya el Pers. ed, Palliser ee +388 

-+40 0 (eG el stot SPN te aiet wha, ho +454 
2 BIU8e A Gelariae 48 8 +402 
3 Oe a Rats Rerauaat: 156 8 +230 
4 ea ea ene ee 22 8 in 
5 — SA inhet: Reha ged x 382 8 —418 





It is evident from the last column that, beginning in a 
cut, the sums increase to the end of the cut, and then dimin- 
ish with the following fill. So the diagram shows a rising 
line for every cut to a maximum which is opposite the grade 
point of the profile; thence a falling line indicates a fill, 
down to its own maximum opposite the next grade point. 

Every loop of the diagram is composed of two equal parts, 
one of cut and the other of fill, and the number of cubic 
yards in each part is indicated by the length of the maxi- 
mum ordinate which divides them, reckoned from any hori- 
zontal line drawn across the loop. The intersection of this 
line with the loop indicates the stations between which so 
much of the cut and its corresponding fill are located, 


263 


HAUL AND THE MASS DIAGRAM 


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264 FIELD ENGINEERING 


So the vertical distance intercepted between any two 
horizontal lines is the measure of the cubic yards of cutting 
on the rising side, or of fill on the falling side, while the longer 
line indicates the extreme haul, and the shorter the least 
haul. Projecting the ends of these lines upon the profile 
will give the corresponding station points. The average 
of the horizontal lines in stations, less the number of stations 
of free haul, multiplied by the number of cubic yards moved, 
gives the product to which the price for “‘ haul’ is to be 
applied. 

In the diagram the loops convex upward indicate the 
haul to be in one direction, while the loops concave upward 
indicate the haul to be in the other direction, since the 
movement is always from cut to fill, that is, from rising 
line to falling line. 

When the grade line is so adjusted that cuts and fills 
balance, and the distances are within practicable limits, 
the zero line is the only line of reference needed. But 
borrowed material, since it appears in the diagram only as 
fill, depresses the diagram, while waste material, appearing 
only as cut, has the contrary effect. In such cases new 
equalizing lines must be drawn, parallel to the zero line 
and as far below or above it as will give an economical 
adjustment. Projecting the ends of such lines upon the 
profile fixes the limits of borrow or waste. 

The most economical distribution of material demands 
that, in making a fill the material should be obtained: 

First, from the nearest cut. 

Second, from a cut such that the cost of haul shall not: 
exceed the cost of excavation. 

Third, from a nearby borrow-pit. 

Borrow should not be made at one point and waste at 
another unless the distance is so great as to make the cost 
of haul prohibitive, or other conditions, such as an un- 
bridged stream, make this course necessary. 

In any case the mass diagram is a valuable aid to a satis- 
factory solution of the question of distribution, as well as 
that of haul. 

292. It is sometimes more profitable to waste material, 
and secure needed amounts from borrow-pits, than to have 
an excessive haul. When, on account of waste of the material 


HAUL AND THE MASS DIAGRAM 265 


from a cut, the embankments have to be built from material 
taken from borrow-pits, the total cost of excavation is 
doubled: This may be denoted by 2c, where c = the cost 
of excavating 1 cubic yard either on the line or in the borrow 
pit. 

When 1 cubic yard is hauled beyond the limit of free haul, : 
the total cost is c + nh, where h = the cost of haul per cubic 
yard per 100 feet and » = the length of excess haul in 
stations. 

Disregarding the cost of the land for the borrow-pit, and 
other considerations, the limit of profitable haul will be 


when 
2c =c+nh 


s OF; 


th 


c 
i (367) 

293. Payment of Overhaul. The following are the 
recommendations of the American Railway Engineering 
Association for all cases where an allowance for overhaul is 
made. eo 

‘““No payment shall be made for hauling material when the 
length of haul does not exceed the limit of free haul, which 
shall be —— feet. 

“The limits of free haul-shall be determined by fixing on 
the profile two points—one on each side of the neutral 
grade point—one in excavation and the other in embank- 
ment, such that the distance between them shall equal the 
specified free-haul limit and the included quantities of ex- 
cavation and embankment balance. All the hau! on material 
beyond this free-haul limit shall be estimated and paid for 
on the basis of the following method of computation, viz.: 

‘All material within this limit of free-haul shall be elimi- 
nated from further consideration. 

‘The distance between the center of gravity of the remain- 
ing mass of excavation and center of gravity of the resulting 
embankment, less the limit of free haul as above described, 
shall be the length of overhaul; and the compensation to 
be rendered therefor shall be determined by multiplying 
the yardage in the remaining mass, as above described, by 
the length of the overhaul. Payment for the same shall 
be by units of 1 cubic yard hauled one hundred (100) feet. 


266 FIELD ENGINEERING 


“Where material is obtained from borrow-pits along the 
embankment and runways are constructed, the haul shall be 
determined by the distance the team necessarily travels. 
The overhaul material thus hauled shall be determined by 
multiplying the vardage so hauled by one half the round 
distance made by the team less the free-haul distance. 
The runways shall be established by the engineer.” 

Referring to Fig. 139, this recommendation may be easily 
explained. The length AB is the free-haul limit. The total 
free haul is represented by the area ACB. The excavation 
in the area BEG is placed in the embankment in the area 
ADF. The number of cubic yards on which overhaul is. 
paid is represented by the ordinate BG, and the total dis- 
tance hauled is given by the line LK. The distance for free 
haul is subtracted from the length of LK, to give the over- 
haul distance. The line LK is drawn through the middle 
point of BG. The centers of gravity of the two portions BEG 
and ADF are considered as at the points L and K. . 

294. Example. ‘The profile and diagram given in Fig. 
139 are taken from an actual line of road. The reasons 
underlying the method of distribution have previously 
been given in § 290. 6090 cubic yards of excavation are 
borrowed between stations 4+ and 9+. The two lines 
DE and EH are “ equalizing lines”’ and give the minimum 
total areas for both the hill and the valley. These lines are 
equal in length. 

Only a portion of the complete problem is shown in the 
figure and this accounts for the borrowed material near the 
right. Other lines of equal length occur beyond the limit 
shown, and it would be unprofitable to haul all material 
forward as far as it would have to be moved. 

The sum of the volumes between successive maximum and 
minimum points should check with the differences of ordi- 
nates of these governing points. 


CHAPTER XVI 


CONSTRUCTION 

295. The engineering department of a railway com- 
pany is usually reorganized for the construction of the road, 
as follows: Chief engineer, Division engineers, Resident 
engineers, Assistant engineers. On some roads the division 
engineers are styled “ Principal Assistants”’; the resident 
engineers, “Assistants”; and the assistant engineers are 
designated according to their duties, as ‘‘ leveler,” ‘“ rod- 
man,” etc. 

A resident engineer has charge of a few miles of line, 
limited to so much as he can personally superintend and 
direct. He has one or more assistants and an axman in his 
party. All instrumental work is done and all measurements 
taken by the resident engineer and his assistants. 

A division engineer has charge of several residencies, 
and inspects the progress of the work on his division once 
or twice a week. In his office, which should be centrally 
located, all maps, profiles, plans, and most of the working 
drawings required on his division are prepared. ‘To him the 
resident engineers make detailed reports once a month, 
or oftener if necessary, which he passes upon as to their 
correctness, and from which he makes up a monthly report, 
or estimate, of the amount and value of the work done and 
materials provided by each contractor on his division. The 
estimates are forwarded about the first of each month to 
the chief engineer, who examines and approves them, re- 
turning for modification any that seem to require it. 

The chief engineer has charge of the entire work, and 
directs the general business of the engineering department. 
He occasionally inspects the work along the line. 

296. Clearing and Grubbing. ‘The first step in the 
work of construction is to clear off al! growth of timber 
within the limits of the right of way. The resident engineer 

267 


268 FIELD ENGINEERING 


with his party passes over the line, making offsets to the 
right and left, and blazing the trees which stand on, or 
just within, the limits o the company’s property. The 
blazed spot is marked with a letter C, as a guide to the con- 
tractor. After felling, the valuable timber should be piled 
near the boundary lines, to be saved as the property of the 
company. The brushwood is burned. 

Where a deep cut is to be made, the stumps are left to be 
removed as the earth is excavated. In very shallow cuts 
and fills the contractor will generally prefer to tear up the 
trees by their roots at once, rather than to grub out the 
stumps after clearing. Where -the embankments will be 
over 3 feet high, grubbing is not necessary; but the trees 
require to be low-chopped, leaving no stump above the 
roots. The engineer should indicate to the contractor 
the localities where each process is suitable. . 

297. While the clearing is in progress, the engineer should 
run a line of test levels touching on all the benches to verify 
their elevations; he may also rerun the center line, replacing 
_any stakes that may have disappeared, and setting guard 
plugs to any important transit points which may not have 
been previously guarded. If any changes in the alinement 
have been ordered, these may be made at the same time. 

298. Cross-sections. The resident engineer is fur- 
nished with a profile of the portion of the line in his charge, 
upon which is plainly indicated by line and figures the 
established grade. The grade given on the profile is that 
which is subsequently called the subgrade, being the surface 
of the roadbed. ‘The final or true grade is the upper surface 
of the ties after the track is laid. 

The methods used in cross-sectioning are fully described 
in Chapter XI. 

The cross-section notes should be traced in ink at the 
first opportunity to secure their permanence. An _ office 
copy should also be made to serve in case of loss or damage 
to the original. 

2939. Shrinkage. In estimating the relative amounts 
of excavation and embankment required, allowance must 
be made for difference in the spaces occupied by the mate- 
rial before excavation and after it is settled in embankment. 
The various earths will be more compact in embankment, 


CONSTRUCTION 269 


rock less so. The difference in volume is called shrinkage in 
the one case, and increase in the other. 

The amount of shrinkage depends on the kind of material 
and the method of placing it in the fill. Dumped from a 
trestle, earth falls so loosely at first that the shrinkage is. 
apparently greater than when the fill is made in layers hauled 
on by wagon or scraper. A trestle fill requires several 
months to Settle, and it is well to give it a little additional 
height to provide for the inevitable settlement. The slope 
stakes, however, are set for the true grade. The surplus 
height, if given at all, is made in the trestle, or is shown by 
poles or stakes driven at 4b from the center line as the. fill 
nears completion. 

But the probable shrinkage must be considered also 

when the grade is first established in order to provide suf- 
ficient material from the cuts, if possible, to complete the 
fills. When the material is paid for as measured in excava- 
tion, the quantity to be placed in the fills is computed only 
for the purpose of properly balancing the work. 
_ Excess of earth allowed for shrinkage is estimated as a 
percentage of the quantity computed from the cross-section 
notes; the excess of height required to provide for shrinkage 
is estimated as a percentage of the height of fill. These two 
-are not necessarily the same. 

The percentages here given are from the average of general 
experience. -They express shrinkage in volume of the several 
classes of materials. 


Pine sand 42200 20% Oat, eae ta), hs og 
Sand and Gravel: .. 8 Surface Soil.<.... 15 
Ordinary Clay..... 10 


When a fill is carried up under traffic, the subsequent 
shrinkage will be less than half the above. —__ 

The percentage in height depends on the height as well as 
on the material. If P is the above tabular per cent of bulk, 
and p the corresponding per cent of height, then p takes the 
following values: 


d Pp d p 
5. feetise snc. Wye sibagie Oe 30 feetcantaipasd .56P 
LO fete 2 Ark bien. wOtle 40ifeet 1. ...untem.. .54P 


BO feeb Get capes WOOL 5O feetzanic,. isle Riva ete 


270 FIELD ENGINEERING 


No slippery earth or quicksand should be allowed in any 
fill. . 

Rock blasted in large fragments increases its bulk about 
60 per cent; in smaller fragments it increases about 75 per 
cent. 

Embankments should be started at full width out to the 
slope stakes, and finished at full width of the roadbed even 
when at a percentage above grade, as there will be both 
lateral and vertical shrinkage. 

Example. A fill of level section, 30 feet. high, is built 
of ordinary clay. Cubic yards per station, 6556; 10 per 
cent extra for shrinkage, 656 cubic yards. Extra height if 
dumped from top, 30 X .10°X .56 = 1.68 feet. Such esti- 
mates are necessarily approximate, being subject to the state 
of the weather, time of year and peculiarities of the soil. 
Frozen embankments settle excessively when thawed out. 

300. Alteration of Line. Inasmuch as the center line 
at grade is the base of reference for all measurements and 
calculations in earthwork, any change made in it after the 
work of grading has begun should be most carefully recorded 
and explained. The center stakes of the old line should 
be left standing until after the new line is established, so that 
the perpendicular offset from the old line to the new, at each 
station, may be measured, as also the distance that the new 
station may be in advance of, or behind the old one. The 
date of the change should be recorded. The original cross- 
sections are extended any amount requisite, the distance 
out being still reckoned from the old center, while a marginal 
note states the amount by which the center has been shifted. 

The difference in length of the lines will make a long or 
short station at the point of closing. The exact length of 
such a station should be recorded, so that it may be observed 
in retracing the line at any time, and in calculating the quan- 
tity of earthwork. The original transit notes of the altered 
line should be preserved, but marked as “ abandoned,”’ 
with a reference to the notes of the new line on another page. 

301. Drains and Culverts. The engineer should 
examine the nature and extent of each depression in the 
profile with reference to the kind of opening required for the 
passage of water. For small springs, and for a limited sur- 
face of rainfall, pipes, in sizes varying from 12 to 24 inches 


CONSTRUCTION Ots 


diameter, serve an excellent purpose as drains. These are 
easily laid down, and if properly bedded, with the earth 
tamped about them, are very permanent; but their upper 
surface should be at least 23 feet below grade. The em- 
bankment is protected at the upper end of the drain by a 
bit of vertical wall, enclosing the end of the pipe. If neces- 
sary, a paved gutter may lead to it. 

Where stone abounds, the bed of a dry ravine may be 
partly filled with loose stone, extending beyond the slopes 
a few feet, which will prevent the accumulation of water. 

When the flow of water is estimated to be too great for two 
lines of the largest pipe, a culvert is required. A pavement 
is laid 1 foot thick, protected by a curb of stone or wood 
‘3 feet deep at each end, and wide enough to allow the walls 
to be built upon it. It should have a uniform slope, usually 
between the limits of 50 to 1 and 100 to 1 to ensure the 
ready flow of water. In firm soils the foundation pit is exca- 
vated 1 foot below the bed of the stream, but if mud is 
found this must be removed and the space filled with rip- 
rap, the upper course of which is arranged to form the pave- 
ment at the proper level. In a V-shaped ravine, requiring 
too much excavation at the sides, and where the fall is 
considerable, riprap may be used to advantage, the bed of 
-the stream above the culvert being graded up by the same 
. material to meet the pavement. In some cases a curtain, 
or cross-wall, is necessary on the lower end to retain the rip- 
rap. 

Culverts should be laid out at right angles to the center 
line whenever practicable, the bed of the stream being 
altered if necessary. The length of an. open culvert is the 
entire distance between slope stakes, the walls being parallel 
throughout, or the length may be taken somewhat less than 
this, and the walls turned at right angles on the upper end, 
forming a facing to the foot of the slope. The walls are 
carried up to grade for the width of the roadbed, and are 
stepped down to suit the slopes. A course is afterwards 
added to retain the ballast. 

In box culverts the span varies from 2 to 5 feet, the height 
in the clear from 2 to 6 feet; the thickness of walls from 
3 to 4 feet; the thickness of cover from 12 to 18 inches, 
and its length at least 2 feet greater than the span. The 


pied FIELD ENGINEERING 


walls terminate in short head-walls built parallel to the 
center line, the top course being a continuation of the cover. 
The length of a head-wall, measured on the outer face, is 
equal to the height of the culvert in the clear multiplied 
by the slope ratio of the embankment. The perpendicular 
distance from the center line to the face cf a head-wall is 
equal to one half the roadbed, plus the depth of the top 
of the wall below grade multiplied by the slope ratio, or 
4b + sk. <A coping is sometimes added. 

302. Arch: culverts are used when the span required 
is more than 5 feet, and the embankment too high to warrant 
carrying the walls up to grade as an open culvert. The 
span varies from 6 to 20 feet; the thickness of arch from 12 
inches to 18 or 20 inches. The height of abutments to the 
springing line varies from 2 to 10 feet, the thickness at the 
springing line from 3 to 5 feet, and at the base from 3 to 6 
feet. The foundations are laid broader and deeper than in 
box culverts, each abutment having its own pit, carried 
to any depth found necessary. The half length of the cul- 
vert is. $b + sk, in which k is the depth of the crown of the 
arch below grade. The abutments are carried up half way 
from the spring to the level of the crown of the arch, and 
thence sloped off toward the crown. The face walls are 
carried up to the crown, and coped. The wing walls stand 
at-an angle of 30° with the axis of the culvert; they receive - 
a batter on the face, and are stepped (or sloped) down to 
suit the embankment. Their thickness, at the base, is the 
same as that of the abutment; at the outer end 3 feet. They 
stop about 3 feet short of the foot of the slope. They need 
not be curved in plan. 

Any stone structure of dimensions pistes than those 
given above, scarcely comes under the head of culverts, and 
should be made the subject of a special design by the engi- 
neer. 

303. Staking out Foundation Pits. For Box 
Culverts. The engineer having decided upon the size 
of culvert required, makes a diagram of it in plan, on a 
page of his masonry book, recording all the dimensions, 
stating the station and plus at which its center is taken, 
the span and height of the opening, etc. He then sets the — 
transit at the center A, Fig, 140, measures the angle between 


CONSTRUCTION ae: 


the center line and axis (making it 90° if practicable); 
on the axis he lays off the distances to the ends of the cul- 
vert and drives stakes at B and C. Perpendicular to BC 
he lays off the half widths of the pit, setting stakes at D and 
£, and laying off DF and FH = AB; and DG and EI = AC. 
On IG produced he lays off CJ = CK, and perpendicular 
to this JM and KJ, and finds the intersections O and N. 
A stake is driven at each angle, and upon it is marked the 
cut required to reach the assumed level for the foundation. 
These cuts are recorded on the corresponding angles of the 








Fre. 140. 


diagram. The pit is thus no larger than the plan of the 
proposed masonry, and the sides are vertical, which answers 
the purpose for shallow pits. 

For Arch Culverts. The pit for each abutment 
when shallow may be of the same dimensions as the lower 
foundation course; if more than 5 feet deep, it should be 
enlarged by an extra space of 1 foot all around. In Fig. 
141 the inside lines show the plan of the abutments at the 
neat lines; the outside lines represent the pits. Having 
prepared a plan of the structure suited to the locality, and 
made a diagram of the same in the masonry book, set the 
transit at A, and drive stakes at D, H, N and O on the center 
line, Then ‘turning to the axis BC,. lay off AC, and set 


274 FIELD ENGINEERING 


stakes at Gand J. With G as a center and a radius equal 
to 2DE, describe on the ground an are cutting HJ in X 
or (IX = DE-cot 30°) may be calculated; and on XG 
produced lay off GK, and perpendicular to this, KL. From 
N lay off NP, parallel to AC, and measure PL as a check. 
Drive a stake at each angle, marked with the proper cut- 
ting, and record the same on the diagram. The locality 
may require the wings to be of different lengths and angles, 
of which the engineer will judge. Guard-plugs should 
be driven in line with the intended face of one or both abut- 
ments, so that the neat-lines can be readily given when 





required. In case the material is not likely to stand verti- 
cally, the pit must be staked out with sloping sides, as 
described below. 

For Bridge Abutments. A design for every impor- 
tant structure is usually prepared in the office after a sur- 
vey of the site. The foundation pit is then laid out from 
dimensions furnished on a tracing, but a diagram of the pit 
should -be made in the masonry book as usual. When the 
bridge is on a tangent, Fig. 142, set the transit at A on the 
center line at its intersection with the axis BC of the abut- 
ment at the level of the seat. Deflect from the tangent the 
angle giving the direction of BC, and lay off AC, AB, setting 
plugs at B and C, and reference plugs (two on each side) 
on BC produced. After staking out the sides of the pit 


CONSTRUCTION 275 


parallel to BC, set the transit at C, and deflect the angle 
for the wing, laying off CD, and driving stakes at the corners 
E and F. Two reference points are then set on the line 
CD produced. The other wing being staked out in the same 
manner, the cut is found at each stake and marked and 
recorded. Cross-sections are then taken near each corner, 
perpendicular to each side, and slope stakes (marked 
‘““ slope ’’) are driven where the slope runs out. Intermediate 
sections are taken when the unevenness of the ground makes 
it necessary, and the lines joining the slope stakes are pro- 
duced to intersect, and other stakes are driven at the inter- 
sections. The position of each stake is shown on the dia- 
_gram, and the cut recorded. 








Hie. 142. 


A slope of 1 to 1 is usually sufficient for pits. If the 
material will not stand at 13 to 1, or if space cannot be 
spared for the slope, the sides may be carried down vertically, 
supported by sheet piling braced from within. 

The reference points should be so chosen that the points 
A, B and C may be found by intersection, on any course 
of the masonry, during the progress of construction. 

When the bridge is on a curve, the bridge-chord 
should be found and the abutments laid out from this. 
Fig. 148. The bridge-chord is a line AB, midway between 
the chord of the curve CD, joining the centers of the abut- 
ments, and a tangent to the curve at the middle point of 
the span. Hence CA = DB = 3MN, which may be laid 
off, and A and B are the true centers of the abutments, 
from which the foundations are staked out as before. 


276 | FIELD ENGINEERING 


The distance CE = DF to the points where the bridge- 
chord cuts the curve is 0.147CD. 

Should an abutment site on a curve be inaccessible, as 
when under water, from any transit point P on the curve, 
lay off PX perpendicular to the tangent at M, observing 
that 

PX = MQ — AC = R (vers PM — 3 vers CM) 
and 
AX = PQ —43AB = R (sin PM — 3CD) 








Fia., 143. 


The point A may then be found by intersection, or by 
direct measurement with a steel tape or wire, driving a long 
stout stake to show the point above the water. Other 
points may then be approximately found, sufficient to 
begin operations. 

In case of a bridge of several spans, the piers are laid out 
in the same manner, from a center point and axis. If ona 
curve, each span has its own bridge-chord, but for conyven- 
ience, the center of a pier may be taken on the center line 
during its construction, and the bridge-chord only found 
for the purpose of placing the bridge; the piers being long 
enough to allow of the shift. 

To locate the centers of piers, a base line is required 


CONSTRUCTION 277 


on one or both shores, and two transits are used to give 
the intersections by calculated angles. When practicable 
the spans should also be measured with a steel tape or wire. 

The bed of a pit for any sort of structure should receive 
the closest scrutiny of the engineer, it. being his duty to 
judge whether the material will resist the load to be imposed 
upon it. <A pit may reauire to be excavated to a greater 
_ depth than first ordered, while sometimes a less depth will 
answer, as when solid rock is found. When a good material 
is reached, if any doubt exist as to its thickness, or as to the 
character of the underlying stratum, borings should be made 
or sounding rods driven down. Piles may be driven to gain 
. the requisite firmness, and a layer of riprap, concrete, or of 
timber may be used to afford a uniform bearing. When 
satisfied of the stability of the bed, the engineer finds the 
the original centers, and gives points for.the courses of 
masonry. A complete record is kept of the amount and kind 
of excavation, the materials used in foundation under the 
masonry, and of the size and thickness of each foundation 
course of masonry; the notes should be taken at the time 
the work is done, it being generally impossible to take 
measurements thereafter. 

304. Cattle-guards are located at highway grade cross- 
“ings. They are placed at right angles across the road at the 
fence lines to prevent the passage of live stock. The sur- 
face guard is the form used most extensively at the present 
time. The material is either wood or metal. 

305. Trestle Work. No wooden culverts should ever 
be used. If stone cannot be had at first, two trestle bents 
may be erected, leaving between them a space sufficient 
to contain the stone structure to be built when the material 
for it can be brought by rail. The bents may be backed 
by plank to retain the embankment, and the stringers are 
then notched down an inch on the caps to receive the 
pressure of the earth, and render the bents mutually sustain- 
ing. The sills are prevented from yielding to the pressure 
of the earth by being sunk in a trench, or by sheet piling. 
Should the span be too long, a central bent may be used, so 
as not to interfere with building the wall. Sometimes pile- 
bents may be used with greater advantage, the piles being 
driven in rows of four each, and capped to receive the 


278 FIELD ENGINEERING 


stringers. In districts where suitable stone is entirely 
wanting, pile or trestle abutments and piers are used for the 
support of bridges, the piles or posts being arranged in groups 
and capped to receive the direct weight of the trusses. They 
should not sustain the embankment, but should be connected 
with it by a short trestle work. 

Trestle work is frequently used as a substitute for embank- 
ment, either to lessen the first cost, or to hasten the com- 
pletion of the line, or for lack of suitable material with which 
to form an embankment. The cost of trestle work, however, 
is not less than that of an earth embankment formed from 
borrow-pits, unless its height exceeds about 15 feet, depend- 
ing on the relative prices of materials and labor. When not 
exceeding 30 feet in height, the bents, for single track, are 
usually composed of two posts, a cap and sill, each 12 X 12, 
and two batter posts, 10 X 12, inclined at 7 to 1, all 
framed together. Two lengths of 3-inch plank are spiked 
on diagonally on opposite sides of the bent as braces. The 
length of the caps should equal the width of the embank- 
ment; the posts should be 5 feet from center to center, 
and the batter posts 2 feet from the posts at the cap. The 
sill should extend about 2 feet beyond the foot of the batter 
post. A masonry foundation for the bent is preferable, 
though pile foundations are not uncommon, and some 
temporary structures are placed directly on a firm soil, 
supported only by mudsills laid crosswise under the sill. 
The spans, or distance between bents, may vary from 12 
to 16 feet. The stringers should consist of at least four 
pieces, two under each rail, bolted together, with packing 
blocks to separate them 2 or 3 inches. Over each bent 
and at the center of each span a piece of thick plank about 
4 feet long should be placed on edge between the two pair 
of beams to preserve the proper distance between them, 
while rods pass through the beams and strain them up to 
the ends of the plank, to increase the stability of the beams 
and prevent their buckling under a load. The stringers 
should be able to carry safely the heaviest load without 
bracing against the posts. The bents, however, if high, 
must be braced against each other. The stringers should 
be continuous, the two pieces breaking joints with each 
other at the bents, to which they are firmly bolted, They 


CONSTRUCTION 279 


may rest directly on the caps, or corbels may intervene. 
The spans on a curve should be shorter than on a tangent. 
The ties should be notched down to fit the stringers closely, 
and guard rails, either wood or iron, secured to them firmly. 
Unless the spans are very short, horizontal bracing should 
be employed, consisting of 3-inch plank, extending from the 
center of each span to the ends of the caps, which are notched 
down to receive the plank. 

For trestles much higher than 30 feet the cluster bent 
is preferable, so termed because each vertical post is com- 
posed of a cluster of four pieces, 8 X 8, standing a little 
apart to allow the horizontal members to pass between 
them. The verticals are continuous, breaking joints, two 
and two, while the horizontals pass the posts and are bolted 
to them at the joints; the framing is accomplished entirely 
by packing blocks and bolts. The batter posts consist 
each of two pieces 8 X 8; the horizontals may be 4 X 10, 
and extend not only across the bent, but from one bent to 
another. Proper bracing is also used in every direction. 
When very high, a secondary pair of batter posts may be 
introduced in the lower part of the structure. The batter 
need not exceed ?th to 1. In some instances two adjoin- 
ing bents are strongly braced together, forming a tower or 
pier, and the piers placed from 50 to 100 feet apart, the 
roadway being carried on trussed bridges. The cluster 
bent admits of any piece being removed and a new one 
inserted when necessary. 

Iron trestles are now adopted where a permanent struc- 
ture is desired. Owing to the expansion of the metal by heat, 
the bents cannot be continuously connected with each cther 
as in a wooden trestle; hence the pier form is resorted to, 
having spans varying from 30 to 150 feet, covered by trussed 
bridges, and the whole structure is more properly styled a 
viaduct. 

306. Tunnels. Tunnels are adopted in certain cases to 
avoid excessive excavations, steep grades, high summits, 
and circuitous routes. Their disadvantages are the in- 
_ ereased time and cost of their construction compared with 
an. open line,,and their lack of light and fresh air when in 
use. It is desirable that they should be on a tangent through- 
out, both for the admission of light and for convenience of 


280 FIELD ENGINEERING 


alinement. Many tunnels, however, have been built with 
a curve at one or both ends or curved throughout.* 

The location of a tunnel, other things being equal, 
should be such as to make not only the tunnel proper, but 
also its immediate approaches by open cut as short as pos- 
sible; and the latter should be selected so as not to be sub- 
ject to overflow, nor lable to landslides. The material to 
be encountered may frequently be determined with toler- 
able accuracy by a study of the geological formation in the 
vicinity, or by actual borings. The most favorable material 
for tunneling is a homogeneous self-supporting rock, devoid 
of springs, which does not disintegrate on exposure to the 
atmosphere. The worst materials are saturated earth and 
quicksand. The presence of water in any material increases 
the cost considerably. 

The alinement of a tunnel is made the subject of special 
survey, after the general location is decided, and this is 
more or less elaborate according to the length of tunnel. 
A permanent station is established at the highest point 
crossed by the tunnel tangent, from which, if possible, 
monuments are set in each direction at points beyond the 
ends of the tunnel. If there are two principal summits, 
stations on these will define the tangent, which may then 
be produced. The monuments established beyond the tun- 
~ nel should be sufficiently distant to afford a perfect back- 
sight from the ends of the tunnel, where other monuments 
are also established. The first qualty of instruments only 
should be used, and these perfectly adjusted, and the ob- 
servations should be repeated many times until it is certain 
that all perceptible errors are eliminated. Since the line of 
collimation will be frequently inclined to the horizon at a 
considerable angle, it is important that it should revolve 
in a vertical plane; and to secure this, a sensitive bubble > 
tube should be attached to the horizontal axis, at right 
angles to the telescope of the transit. The distance may be 
obtained by triangulation, though direct measurement is 
to be preferred. A steel tape is convenient and accurate, 
providing that allowance be made for variations due to 
temperature, from an assumed standard. The rods de-— 


* The Mont Cenis tunnel, requiring a curve at each end, was first 
opened on the tangent produced, giving a straight line through, and 
the curves were excavated subsequently. 


CONSTRUCTION 281 


scribed in § 43 may be used instead of plumb lines, the tape 
being held at right angles to them, and therefore horizontal. 
A plug should be driven for each rod to stand on, and a center 
set to indicate the line and measurement. 

As the excavation of the tunnel proceeds, the center line 
is given at short intervals by points either on the floor or 
roof. Overhead points are generally preferred, from which 
short plumb lines may be hung, constantly indicating the 
line, with little danger of being disturbed. When a new 
transit point is required in the tunnel, it should be estab- 
lished directly under an overhead point, which serves 
as a check upon its permanence, and as a backsight when 
needed. 

Shafts are sometimes opened to give access to several 
points of the tunnel at the same time, thus facilitating the 
work, though at an increased cost. They also serve for 
ventilation during the progress of the work, though they 
are worse than useless for this purpose afterward, except 
possibly in the case of a single shaft near the center of the 
tunnel. Some of the longest tunnels have been formed 
without shafts, while many shorter ones have had several, 
whith have generally been closed after the tunnel was com- 
pleted. Shafts are either vertical, inclined, or nearly hori- 
_ zontal; in the latter case they are called adits. Inclined 
shafts should make an angle of at least 60° with the vert- 
ical. Vertical shafts may be either rectangular, round, or 
oval. ‘Their dimensions vary, depending on their depth and 
the material encountered, between 8 and 25 feet. They are 
usually sunk on the center line of the tunnel, though some- 
times at one side. When over the tunnel the alinement 
below is obtained directly from two plumb lines of fine wire 
suspended on opposite sides of the shaft from poimts very 
carefully determined at the surface. The, plummets are 
suspended in water to lessen their vibrations, and as soon as 
the transit can be set up at a sufficient distance to bring 
the lines into focus, it is shifted by trial into exact line with 
the mean of their oscillations, the latter being very limited. 
Permanent points may then be set, but should be repeatedly 
verified. As soon as the workings from a shaft communi- 
' eate with those from either end, or from another shaft, the 
alinement thus found is tested, and revised if necessary, 


/ 


282 FIELD ENGINEERING 


These operations require the greatest nicety of observation 
and delicacy of manipulation to obtain satisfactory results. 

From plumb lines in the central shaft of the Hoosac tun- 
nel, the line was produced three tenths of a mile, and met 
the line produced 2.1 miles from the west end with an error 
in offset of five sixteenths of an inch. In the Mont Cenis 
tunnel the lines met from opposite ends with “no appre- 
ciable’”’ error in alinement, while the error in measurement 
was about 45 feet in a total length of 7.6 miles. 

When a curve occurs in a tunnel it is usually near one 
end. The tunnel tangent is produced and established as 
before described, and a second tangent from some point 
on the curve outside the tunnel is produced to intersect it, 
the intersection being precisely determined and the angle 
measured with many repetitions. The tangent distances 
are then calculated, and the position of the tangent points 
corrected by precise measurements, and permanent monu- 
ments are established. As the tunnel advances, points may 
beset at short intervals on the curve in the usual manner; 
but at intervals of 100 feet the regular stations should be 
defined with finely centered. monuments, using a 100-foot 
steel tape carefully supported in a horizontal position. 
When it is necessary to use a subchord, its exact length should 
be calculated as shown in § 107. When the curve has 
advanced so far as to render a new transit point necessary, 
this should be established at a full station. The subtangents 
from the two transit points should then be produced to inter- 
sect, and measured for equality with each other and with 
their calculated length. The distance from their intersec- 
tion to the middle of the long chord should also be measured 
as a check on the deflections. When no perceptible errors 
remain, the curve may be produced as before until the P.T. 
is reached. It,is evident that correct measure is indispens- 
able to correct alinement on curves. Should obstacles on 
the surface necessitate triangulation, more than ordinary 
care must be exercised, and as many checks introduced as 
possible. The triangles should be so arranged that all of the 
angles and most of the sides may be measured. 

Test levels are carried over the surface with great care, 
each turning point being made a permanent bench, and 
its elevation determined with a probable error not exceeding 


CONSTRUCTION 283 


0.005 foot. Levels may be carried down a shaft on a series 
or bolts or spikes about 12 feet apart in the same vertical 
line, the distances being measured by the same _level-rod 
as that with which the benches are determined. The 
measures should be taken between two graduations of the 
rod, not using the end of the rod, which may be slightly 
worn. Fine horizontal lines on the heads of the bolts may 
be used to mark the exact distances. After the shaft reaches 
the level of the tunnel, the depth may be measured more 
directly with a steel tape, the entire length of which has been 
corrected at the given temperature, by comparison with 
the same rod. 

If the grade of a tunnel is to be continuous, it should 
be assumed at something less than the maximum of the road, 
but not less than 0.10 per station, which is required for 
drainage. If a summit is to be made in the tunnel, the 
grade from the upper end should not exceed 0.10 per station. 
Grades are given in the tunnel from day to day, or as often 
as required by the progress of the work, the marks being 
made on the sides at some arbitrary distance above grade. 
Turning points should be taken on permanent benches. 

The least width of a tunnel in the clear should be, for 
single track about 16 feet, and for double track this distance 
should be increased by the distance between tracks. The 
height in the clear above the tie should be 22.5 feet for 
both single and double track, allowing for tie and ballast, 
this distance being measured at the center lines of each 
track. The form of section depends somewhat on the 
material traversed. In perfectly solid rock a nearly rec- 
tangular section may be used, the roof being slightly rounded. 
In dry clay, and stratified rock, a flat arch may be used, 
and in other cases a full-centered arch. The latter form is 
rather to be preferred on account of the better ventilation 
afforded. The sides are made vertical, battered or curved, 
as necessity or taste may dictate. In wet and infirm soil 
an invert floor may be required, otherwise it is made level 
transversely. When a lining is required the original section 
must of course be made large enough to allow for the masonry, 
and the temporary timber supports behind it. Hard-burned 
brick is usually adopted for arching, being durable and easily 
handled. .In loose rock the arching may be from 13 to 26 


284 FIELD ENGINEERING 


inches thick, in wet and yielding soil a thickness of from 
26 to 39 inches may be necessary. The walls may be from 
2% to 6 feet thick. 

In forming a tunnel, a heading or gallery of smaller 
cross-section is first driven and afterward enlarged to the 
full size required. In firm clay or loose rock which will 
temporarily support itself until the masonry can be put 
in, it is better to drive the heading along the floor (at sub- 
grade) of the tunnel, the remaining material being then 
easily thrown down in sections as the arching is advanced. 
In solid rock, or wet earth, a top-heading (along the roof) is 
generally preferred. The dimensions of a heading driven 
by hand are usually 8 feet high by 8 or 10 feet wide, but in 
solid rock where drilling machinery is introduced, it is ad- 
vantageous to make the heading as wide as the tunnel at 
once. By drilling holes into the face at points about 5 feet 
each side of the center, and converging on the center line at 
a depth of about 10 feet, a triangular mass of rock may be 
blown out, and the space thus gained facilitates the blast- 
ing of the adjacent rock on either side. An advance of about 
10 feet in each day of 24 working hours may thus be made, 
using nitroglycerine in some form as the explosive agent. 
Owing, however, to unavoidable delays from various causes, 
this rate or progress cannot always be maintained. At the 
Hoosac tunnel the greatest advance in one week was 50 feet; 
in one month 184 feet at one heading. At the Musconet- 
cong tunnel a heading 8 X 22 feet in syneitic gneiss was 
advanced at the average rate of 137 feet per month for 
six months, the maximum being 144 feet—the enlargement 
of the tunnel to fuli size going on at the same time, a few 
hundred feet behind. At the St. Gothard tunnel the north 
heading 2.5 X 3 meters was advanced in mica gneiss, during 
the year 1875 at the average daily rate of 3.71 meters, with 
a maximum of about 4 meters, but the enlargement was not 
made. The south heading advanced at the rate of 2 meters 
a day, timbering being at times necessary. 

In ordinary clay a heading may be driven at from 75 to 
180 feet per month, according to circumstances, where 
timbering is put in. The enlargement, including timbering 
and masonry, may be advanced at from 20 to 60 feet per 
month. Small tunnels for water conduits are driven through 


- CONSTRUCTION 285 


dry clay at the rate of 10 feet per day, the masonry following 
at once without timbering. 

The compressed air used to drive the drilling machinery 
serves to supply ventilation also. When this is wanting or 
proves insufficient, exhaust fans are used. At Mont Cenis 
a horizontal braitice or partition was built in the tunnel, 
dividing it so as to secure a circulation of air. When foul 
gases are encountered, ventilation becomes a serious ques- 
{ion, and in one instance an important work was abandoned 
for this cause. 1 

Cross=sections of the heading, and also of the tunnel 
enlargement, should be measured at intervals of about 20 
. feet, as soon as opened, to see that the sides, roof, and floor 
are taken out to the prescribed lines, at the same time that 
the latter are exceeded as little as pcssible. In solid rock, 
since some material outside of the true section will neces- 
sarily be thrown down, leaving an irregular outline, it is 
well to take two cross-sections at the same point, one fol- 
lowing the projections and the other the recesses of the rock, 
from which an average section may be estimated. A daily, 
or at least a weekly, record of operations should be kept 
in tabular form, and the progress indicated by a profile and 
cross-sections drawn on a sufficiently large scale to show 
details. 

The drainage of a tunnel is best secured by a line of 
stoneware or cement pipe laid in a trench along each side, 
and covered with ballast or other loose material. The entire 
floor is thus made available for the use of the trackmen. 
When an invert is-used, the drain is placed in the center 
between tracks. If the amount of water is large, drain 
pipe may be laid behind the walls, and the back of the arch 
may be covered with asphaltum, or coal tar, to prevent — 
a censtant dripping on the track. 

307. Retracing the Line. As the grading progresses, 
in either excavation or embankment, the principal transit 
points are established on the roadbed from the points 
of reference, and the center line is retraced, setting stakes at 
every 50 feet. Transit pomts on grade’should be fixed upon 
stout, durable posts firmly set in the ground, and standing 
high enough to be easily reached after the ballast is laid. 
To recover the old line, any discrepancies in measurement 


286 FIELD ENGINEERING 


must be left between the transit points where they occur, 
and not carried forward. In retracing a curve, if the transit 
is placed at the forward point, allowing the chain to advance 
toward it, slight differences in measurement will not affect 
the position of the curve. If any short or long stations have 
been introduced on the location, their position on the line 
must not be changed in retracing. The chain may be 
adjusted so that its measures will agree with the recorded 
distances between transit points. Offsets are made right 
and left from the new stakes to see that the roadbed is of 
the full width at all points. The levels are also carried over 
the grade, and any remaining cut or fill found necessary 
is marked on the back of the stakes, due to allowance being 
made for the probable settlement of embankments. 

308. As the work approaches completion the contractor 
goes over the line, dressing it to grade and opening the side 
ditches if this has not been previously done. 

Drain-tile should be laid at the bottom of these ditches 
and lightly covered with earth, particularly if the cut be wet. 
These not only prevent the water from reaching the ballast, . 
but by keeping the foot of the slope comparatively dry pre- 
vent the earth from sliding down and filling up the cut. 
There is also a marked economy in their use, as the cost is 
trifling, and all further excavation of mud and water from 
the cut is generally.obviated. Should any springs appear 
in the slope a branch line of smaller tile may be laid to meet 
it. If the slope is liable to be overflowed from the surface 
above, an open ditch should be dug a few feet beyond the 
slope stakes, leading the surface water to discharge else- 
where. 

309. The final estimate is a complete statement in 
detail, of the amount of work done and materials provided, 
in the construction of the road, and is the basis of final 
settlement between the company and contractor. Its prep- 
aration should be begun as soon as possible after the work 
is in progress, and should be continued, as fast as the neces- 
sary data are accumulated, while the circumstances are 
still fresh in mind, and when any. omissions in the field 
notes may be readily supplied. The content of each pris- 
moid, the classification of its material, and the length of haul 
to which it is subject, should be matters of special record 


CONSTRUCTION 287 


in a book provided for that purpose. These results having 
been carefully computed by exact methods, form a standard 
of comparison for those approximate results which must 
be had from time to time during the progress of the work, 
and furnish a limit to the amounts of the monthly estimates. 
The same remark applies to all other items of labor and 
material. The notes and record of the final estimate should 
be particularly full and exact in respect to all such items as 
will be inaccessible to measurement at the completion of 
the work, such as foundation pits, foundation courses of 
masonry, culverts, and works under water. 

310. Monthly Estimates. On or before the last day 
of every month during the progress of construction, measure- 
~ ments are taken to determine the total amount of work done 
and material provided up to that date. The estimates 
based on these measurements are called Monthly Estimates. 
It is frequently necessary to take measurements for both 
monthly and final estimates at other times than the end 
of the month, as in the case of foundations which are not 
long accessible. With respect to each piece of work satis- 
factorily completed, the monthly estimate should be exact, 
and identical in amount with the final estimate. With 
respect, however, to items of work in progress at the time 
of measurement, the monthly estimate is only approximate, 
yet should be as precise as the nature of the case will allow; 
and the quantities stated should not be in excess of fair 
proportion of the total quantities given on the final estimate 
for the same piece of work. 

A special field book is devoted to monthly estimate 
notes. Each page should be dated with the day on which 
the notes upon it were taken. The notes consist of measure- 
ments of all sorts, principally of cross-sections partially 
excavated. These sections should be at the same points 
on the line as the original sections, so that comparisons 
may be made. Wherever the excavation is finished to 
grade, it is only necessary to write ‘‘ completed ”’ opposite 
such stations, and the quantities may be taken from the 
final estimate or computed from the original notes. It is 
frequently necessary to retrace portions of the center line in 
taking estimate notes, so that all the field instruments are re- 
quired, but a party of three or four men js usually sufficient, 


288 FIELD ENGINEERING 


If the contractor has provided materials, such as stone, 
lumber, etc., which are not as yet put into any structure 
when the estimate is taken, these should be measured and 
entered under the head of temporary allowance, an 
arbitrary price being used somewhat below the actual value 
of the material as delivered. Such allowances should never 
be eopied from one month’s estimate to the next, but made 
anew on such material as may be found that seems to require 
it. But all completed items of contract work, and of extra 
work when ordered by the engineer, are necessarily copied 
from one nionthly estimate to the next during the contin- 
uance of the contract. 

A blank form is used by the resident engineer in report- 
ing monthly estimates, on which a column is provided for 
each class of material and work required by the contract, 
while the several lines, headed by the numbers of the proper 
stations, are devoted to the different cuttings, structures, 
ete., In consecutive order as they occur on the line of road. 
The estimates are made out and reported separately for the 
several sections into which the line of road is divided for 
letting. 

These reports are reviewed by the division engineer, 
and the footings copied upon another blank, which is the 
monthly estimate proper; the prices are attached to the 
items, and the amounts extended and summed up. This- 
sum indicates approximately the total amount earned by 
the contractor up to date, from which is deducted a certain 
percentage (usually 15 per cent), which is retained by the 
company until the completion of the contract. From the 
remainder is deducted the amount of previous payments, 
which leaves the amount due the contractor on the present 
estimate. A blank form of receipt is appended, to be 
signed by the contractor, 


CHAPTER XVII 
TRACK LAYING 


311. After the roadbed has been prepared, the ties and 
rails are laid to approximate line, thus enabling the ballast 
train to enter. Ballast is deposited in layers of about 6 to 
8 inches and the track is gradually raised and brought into 
- position, both horizontally and vertically. To secure this 
two sets of stakes are used; the center line stakes, and the 
grade stakes. _ 

312. Center Line Stakes. On long tangents, one stake 
in every 200 feet is sufficient; on ordinary curves, one stake 
in every 50 or 100 feet, and on very sharp curves one in every 
25 feet. Having located the center stakes, the rails are 
brought into position by use of the track gauge. 

The center point of the track gauge is placed at the tack 
in a stake, and the gauge sides of the rails are brought into 
contact with the lugs at the ends of the gauge. The rails at 
other stakes are similarly adjusted and the track between 
successive points is lined up by eye, whether on tangent or 
curve. 

313. Curving Rails. Before any rail is spiked to its 
place in a curve, it must be evenly bent from end to end, 
so that it will assume the proper curvature when lying free. 
The bending may be done by using sledges, but is best 
accomplished, especially for turnouts ard other sharp curves, 
by using a bending machine made especially for this purpose. 

The proper ordinates for locating or bending curved rails 

may be found in different ways, of which some of the more 
~ usual will be given in the next two sections. 

314. To find the middle ordinate m, for 1. station, 
or 100 feet, on any curve, in terms of the degree of curve D. 

Referring to Fig. 144 we have in the right triangle AGH 


GH = GA-tan GAH 
289 


290 FIELD ENGINEERING 


But GA = 4AB = iC, and (Table XLI, 28) GAH = +AOB = 
ZA; hence 

M = 3C-tan 7A (368) 
a general expression for the middle-ordinate of any chord. 


If in this equation we make C = 100, A becomes D; and 
denoting the corresponding value of M by m, we have 


m = 4100 tan 4D (369) 


whence the rule, Multiply the nat. tangent of 4 the degree of 
curve by 100 and divide by 2. Thus the values of m in the 
5th column of Table I have been calculated. 





Fig. 144. 


315. To find the middle ordinate for any chord in terms 
of the chord and radius. 
Referring to Fig. 144, we have 


GH = OF - OG = OF — VAO? — GA? 
or 


eee 
sent Nic a (5) (370) 
When C = 100 we have for the middle-ordinate of one 


station 
m= R — VR? — 2500 (371) 


For any subchord ¢, less than 100, we have for the middle 
ordinate, 


TRACK LAYING 291 


m = RB — a/R? Fs 


or (372) 
m = Rk — \ (2 +8) (x - 5) 


4 
By adding Bins to the quantity under the radical in eq. 


(872) it becomes a perfect square, giving 
22 


My cE nearly, (373) 
which is a very useful formula, although approximate. The 
error in m, does not exceed .002 for any subchord c when 
the radius is greater than SCO. On a 20° curve the error 
will be .002 for a chord of 50 feet; and on a 40° curve the 
error in m, will be only .003 for a chord of 33 feet. Equation 
(373) is therefore practically correct in all cases for finding 
the middle-ordinates of rails. Table VIII is calculated by 
eq. (372). 

From the last equation it appears that, with a given 
radius, the middle ordinate varies nearly as the square of 
the chord. We may therefore find the middle ordinate 
of a rail whose length is c by the proportion 

(LOO) foc? cates Wig 
or, 


10000 
in which m is obtained from Table I, column 5, for the given 
radius of degree of ‘curve. 

Example.—What is the middle ordinate of a 33-foot rail 
when curved for a 20° curve? 


1089 X 4.374 
10090 


When a long rail is bent for a sharp curve, observe that c 
is the length of the chord of the rail—not of the rail itself. 
For the chord of half a rail the middle ordinate is one- 
fourth the middle-ordinate of the whole rail. Thus, in the 
above examplé it would be 0.119 or 17% inches. 
Instead of using the chord of the whole rail, it may be more 
convenient to assume a chord shorter than the rail, especially 


nearly, (374) 


hi == 


Eq. (874) . m = = 0.476'= 54241n. 


292 FIELD ENGINEERING 


when the chord is not an exact number of feet, knotting the 
string to the length assumed, and applying it to different 
portions of the rail successively. 

316. Testing Track Curvature. The proper curva- 
ture of a rail is tested by measuring its middle-ordinate 
from a small cord stretched from end to end and touching 
the side of the rail-head. The cord should also be stretched 
from the middle point of the rail to either end, and the 
middle-ordinate of each half length measured, to test the 
uniformity of curvature. 

A common rule for finding the degree of curve of a track 
is “ The middle-ordinate in inches of a 62-foot chord equals 
the degree of curve.’ This follows from eq. (373) by sub- 











Sr 1s 


S 


oY 
y 
4 
Z7 
o<---=-f-<-- 
~ 2 & 
“SA 
NS 
~~ 


ec 
~ 


BiGs tS: 


stituting 1 inch for m, and 5730 for the radius of a 1° curve. 
Assuming that the radius varies directly as the degree of 
curve, the rule obtains as stated. 

Another method is illustrated by Fig. 145. 

Select a point A at a rail joint; sight along a fine tangent 
to the inner rail at a point as D, and note its intersection. 
with the other rail at C. Count the number of rail lengths 
whole and fractional in ABC. 

Let » denote the number of rail lengths in ABC, 


5730 


Combining eq. (3873) and Rk = D” and solving for D 


we get 





ae 8 X ae 


C 


TRACK LAYING 293 


Considering the lengths of the chord AC and the are ABC 
equal, we have for 30-foot rails, c = 30n, and for 33 -foot 
rails ¢ = 33n. 

This equation for the standard gauge of 4 feet 83 inches = 
4.708 feet becomes for 30-foot rails 


9) . 
a = (approx.) (375) 
and for 33-foot rails 
198 "9 
D= a. (approx. ) (376) 


Equations (375) and (876) give the degree of the outer 
rail, and that only roughly, depending on the actual gauge, 
which is sometimes increased on curves. 

317.. Grade stakes are placed at one side of the track. 
‘The tops of the stakes are set for the grade of the top of the 
rail, which is at a certain definite height above the ‘“ sub- 
grade” of the profile and cross-sections. 

Track on tangents is level transversely. If the grade is 
uniform, the elevation of stations differ by a constant amount. 
On curves, the top of the stake indicates the elevation for 
the outer rail, the inner rail carrying the regular grade. 

The track is gradually raised by track jacks and by tamp- 
ing until it is at the proper elevation, as proved by a track 
level extending from the top of a stake to the top of the 
rail. 

318. Vertical Curves. The established grades of a 
road appear on -the profile as right lines which are either 
level or else so inclined as to have a definite rate of rise per 
station, and as a station measures 100 feet, this rate is often 
described as a percentage, a grade rising 1 foot per station 
being known as a | per cent grade, for instance. The chosen 
grade must not only have a definite rate, but it must termi- 
nate at a station (rarely at a half-station) where it meets 
another right-line grade and makes with it a definite angle. 
This angle, however, is not estimated in degrees, but by the 
difference in the rates of the grades. 

If the grades—both of them rising—have the respective 
rates of g and g’, the angle is expressed by (g — g’). Grades 
rising in the direction in which the survey was made are 
marked and called +, and grades falling in the same direc- 
tion are called —, but the above expression is always true 


294 FIELD ENGINEERING 


algebraically. If both grades were —, their difference would 
still define the angle, but if one were rising and the other 
falling, then the sum of the rates, either plus or minus, would 
define the angle between the grades. 

For the. safety and convenience of the traffic this angle 
must be replaced by a vertical curve, tangent to the grade 
lines, the total angle being distributed among its chords. 
The parabola is always used for vertical curves, and the 
tangent points must be fixed at equal distances from the 
vertex so that the diameter through the vertex, and all 
offsets parallel to it, may, be vertical. The curve must be 
long enough to make the transition easy. As heavy trains 
reach the top of a long grade at a lower speed than that 
with which they reach its foot, it is customary to allow much 
shorter curves at summits than at sags. ; 

To determine the length of a vertical curve, let » be the 
number of stations in the curve, and 2a the linear deflection 
from chord to chord thought desirable. ‘Then 


nee (377) 


If this value of n is fractional, take the nearest even number 
(preferably greater), as the final value of n. Then by 
inversion, 


LOG ) 
eae (378) 


which is the linear deflection from the grade tangent to the 
first station on the curve at either end, since a tangent 
deflection is one half the chord deflection. 

The values of a and n being decided, it is a simple arith- 
metical operation to tabulate the elevations of the stations 
of the curve, as will appear from the following: 

319. Hxample-—Fig. 146. Given, a rising grade of 0.8 
to station V; thence a falling grade of 0.4; to join the grades 
with a vertical curve in which 2a = 0.20, and let the eleva- 
tion of V be 117.0 feet. 

By eq. (377) 


bans AYO BO ert Cato oe 6, and each tangent = 3. 


0.20 
Then _ 
117.0 — 3 X 0.8 = 114.6 = elevation of sta. A, 


and 
117.0 —3 K 0.4 = 115.8 = elevation of sta, C, 


TRACK LAYING ; 295 


Calling sta. A zero, and numbering the curve stations 
from 0 to 6, we prepare the following statement: 


Sta. Defi. Cor. Rate Elev. Sta. 
0 0 +.8 114.6 A 
1 a —.1 +.7 115.3 Hae 
2 2a — 2 +.5 115.8 L’ 
3 2a —.2 +.3 116.1 B 
4 2a — .2 +.) 116.2 M’ 
5 2a —.2 —.1 116.1 N’ 
6 2a — .2 —.3 115.8 C 
a —.1 —.4 tangent, 





Frei 146. 


The elevation of each station is obtained from the pre- 
ceding after correcting the rate, and the last rate would give 
the elevation of the next station on the grade beyond C. 
This, and the elevation of C proves the statement correct. 

Had the first grade been a descending one its rate would 
have been minus-and the corrections plus. Otherwise the 
operation would be the same. The elevations are the grades 
at the several stations by which grade stakes are set, either 
for the original cross-sectioning or for final test. 

If it is desired to set grade stakes every 50 feet we have 
but to double the value of n and halve the value of g and of 
a, and proceed as before. 

This solves the whole problem; but there are other ways 
of arriving at the same results, which will be noticed briefly, 
as they are preferred by some. 

320. It is a property of the parabola that offsets from 
tangent to curve are to each other as the square of their 
' distance from the tangent point. Hence, if we call A zero 


296 FIELD ENGINEERING 


and number the stations in order up the grade tangent, we 
shall have in the example given above: 























l 
Stationh:.~ aa che A hos Ler esi? 3 4 5 6 
Offsets. Foxy ea 0 | a | 4a 9a l6a | 25a 36a 
Offset te wee Se Se ai 2. SelyaGe 6 
Tan. Grade 114.6 | 115.4 | 116.2 | 117.0 | 117.8 | 118.6 | 119.2 
Curve Grade...... 114.6 | 115.3 |.115.8 | 116-1 | 116:2 | 116.1] 1 
Station: ran ae wee A K’ L’ B mM’ |. N’ C 























In this method the grade tangent is produced from V to 
G and the last offset is represented in the figure by the line 
GC, which is just four times the length of VB. 

This method may be somewhat simplified, especially on a 
very long curve, by observing that the figure is symmetrical 





Fig. 147, 


about the diameter VH drawn through the vertex, so that 
MM’ = LL’, etc. Therefore, after the point V we may 
write the elevations of the succeeding stations on the second 
tangent and apply to them the offsets already obtained, but 
in reverse order. The result is the same. 

321. A third method of computing grade elevations on 
vertical curves is by means of a vertical ordinate from the 
long chord, AC, Fig. 147. The elevation of the middle 
point of this chord is a mean of the elevations of grade at 
A and C. Thus we obtain VD, which the parabola bisects 
at B, giving us the middle-ordinate, DB = VB, or 


grade A + grade 4 
a 





VB.=3 (rade V— (379) 

Divide AV into any number of equal parts and divide 
VB by the square of the number, and the quotient is the 
vertical offset from tangent to-curve at the first point from 
A. Other offsets are proportional to the square of their ° 


TRACK LAYING 297 


distance from A as in the former case. Treat VC in the 
same manner; the offsets will be the same for the same 
number of points. 

322. As to desirable values for the linear deflection, 
2a, the Am. Ry. Eng. Association has recommended the 
limits of 0.10 on summits, and 0.05 in sags for first-class 
roads with heavy traffic, while for minor roads with light 
traffic these limits may safely be doubled. It would be well 
to change the above limit of 0.05 to 0.06 in order to avoid 
the third decimal place in figuring grades. 

323. HKlevation of the Outer Rail on Curves. 
When a car passes around a curve, a centrifugal force is 
developed which presses the flanges of the wheels against 
the outer rail. This feree acts horizontally, and varies 





ts f 
Fie. 148. 


as the square of the velocity, and inversely as the radius of 
the curve. Denoting the centrifugal force by f, we have 
Dewy 
32.166P’ 
weight of loaded car in pounds, v = velocity in feet per 
second, and Rk = radius of curve in feet. 

In Fig. 148 let ab represent a level line at right angles 
to the track, let a and c be the tops of rails on a curve, let 
be =e = elevation of outer rail c, and let the point d be 
the center of gravity of the car. The foree f acts in the direc- 
tion ab, and if f’ = the component of f in the direction ac, 
then 


from the theory of mechanics f = in which w = 


fsa feeaboti Gc 
The weight w, resting on the inclined plane ac develops 
a component in the direction ca, and denoting this by w’, 
we have by similar triangles, 
. ap ap? bee ae: 


298 FIELD ENGINEERING 


Since equilibrium requires that w’ shall equal f’, we have 


J Ge 


after dividing one proportion by the other iter 


f= on Equating this value of f with that given above 
we find, 
4, 0D.20" 
* ~ 32.166R 
But ab =.V ac? — e*, and ac = distance between rail cen- 
. 2 
ters = gauge + onerail head = g + 0.188. Alsov = —V, 


if V denotes the velocity in miles per hour. Making these 
substitutions and reducing, we have 
V2 


06688 


is : po 
\ La (.06688 5) 


By this formula Table IX is calculated for the standard 
gauge g = 4’ 83", = 4.708. 
An approximate formula may be obtained by assuming 


(380) 





e = (g + .188) 





that ab = g for practicable values of e. Substituting this 
in the first value of e given above, and replacing v by epee te 
we have 

gV* ; 
(approx.) ~ osessa— (381) 


R 


which is the formula generally employed. 

In laying a new track, the transverse inclination is first 
given to the ballast by grade pegs driven either side of the 
center line at a distance of (g + .188) each side of the center; 
the outside peg being set higher, and the inside peg lower 
than the grade of ballast on the center line, by the proper 
elevation selected from Table IX. But in resurfacing an 
old track, the inner rail is taken as grade and the outer 
rail is raised the necessary amount. 

324. The proper elevation may be found mechan- 
ically by the following method: 

To find on a curved track, the length of a chord whose middle- 


TRACK LAYING 299 


ordinate shall equal the proper elevation of the outer rail for 
any velocity V in miles per hour. 

By the conditions of the problem, we have m in eq. (373) 
equal to e in eq. (881), or 


ct _ gV?.06688 


Sk R 
Ee = .73144VVg (382) 
When g = 4.708, 
c = 1.587V (383) 


Lay off the chord, c, upon the rail of the track, stretch a 
piece of twine between the points so found, and measure the 
- middle-ordinate; it will equal the proper elevation. 

325. The velocity assumed in the preceding formulas 
should be that of the fastest regular trains which will pass 
over the curve in question, since the flanges would be forced 
against the outer rail were there no-centrifugal force devel- 
oped, by reason of the wheels being rigidly attached to the 
axles, and the axles being parallel. 

ey He rails on tangents should be level transversely except 
on very flat curves where spirals are not employed. In 
such cases one rail of the tangent is gradually elevated, 
beginning a short distance from the P.C. and the P.T. 
At a P.C.C._the elevation should be an average of the ele- 
vations due to the two ares. Owing to the difficulty of prop- 
erly adjusting the elevation of rail, it is objectionable to have 
ares of very dissimilar radii join each other; and the ob- 
jection is much greater in the case of reversed curves unless 
separated by a short tangent. See § 82. 

On the other hand, a short tangent between arcs which 
curve in the same direction should be avoided, since it 
makes a “ flat place ’’ both in line and levels, at once un- 
sightly and injurious to the rolling stock. 

In the case of turnouts, however, no elevation of rail is 
possible (except when both tracks curve in the same direc- 
tion). . 

326. Owing to the expansion of the rails by heat, 
a space must be left at the rail-joints. The highest tem- 
perature of a rail in the summer sun is about 130° Fahr. The 
expansion of iron and steel per 100° is .0007 per foot; or for 
a 33-foot rail .023 foot or .276 inch, Therefore when 33- 


300 FIELD ENGINEERING 


foot rails are laid at a temperature near the freezing-point, 
or 100° below the maximum, the space allowed must be at 
least a quarter of an inch. At 80° Fahr. or 50° below the 
maximum, it need be only half as much. The space required 
is also proportional to the length of rail used. The exact 
space should be given, as less would result in the rails being 
forced up by expansion, while more than necessary space 
gives a rough road, and hastens the destruction of the rail. 
Wherever sidings are required, the necessary frogs and 
long switch-ties should be provided in advance, so that they 
may be put in place at the time of laying the main track. 
For every road crossing at grade, heavy oak plank should 
be provided, and laid upon the ties as soon as the rails are 
spiked, so that the highway travel may not be impeded. 


CHAPTER XVIII 


TOPOGRAPHICAL SKETCHING 


327. Topographical sketches taken on preliminary sur- 
veys are usually of great value in projecting a line for 
location; they should be made therefore as accurate and 
complete as possible. In too many instances sketches are 
presented having a picturesque appearance, but conveying 
. little information, if not tending to mislead the map-maker. 
The aim of the topographer should be to record the topo- 
graphical features either side of the line with as much pre- 
cision as those directly upon the line, without taking actual 
measurements, except in rare instances. The eye and the 
judgment must be usually depended on for distances and 
dimensions. The sketch of a tract extending to 400 feet 
each side of the line ought to be accurate enough to warrant 
its being copied literally upon the map. If a much wider 
range is required it may be advisable to use the plane-table; 
but an approximation to plane-table methods may be em- 
- ployed in ordinary sketching. 

328. As artificial features are the most readily de- 
fined and located, these should first receive attention in 
making a sketch. When recorded they form a skeleton 
upon which the natural features can be drawn with more 
precision than if the order were reversed. The point where 
each fence crosses the line and the angle between the two 
may be sketched exactly. The distance along the fence to 
any object may be estimated, and checked (in case of an 
oblique angle) by observing where a line from the object 
perpendicular to the center line would intersect the latter. 
The book may be rested on any support, the center line 
of the page coinciding with the line of survey, and the direc- 
tion of objects defined by a small ruler laid on the page. 
This operation being repeated from another point gives 
intersections which locate the several objects on the sketch. 

301 


302 FIELD ENGINEERING 


If the bearings are taken they may be plotted on the page 
as well as recorded, giving the same results. The eye may 


be trained to estimate distances correctly by observing the _ 


appearance of objects along the measured line, the distances 
to which are therefore known. 

329. After the artificial objects the more distinct natural 
features are to be sketched, as streams, shores, margins of 
swamps, forests, etc., ravines, ridges, and bluffs, taking care 
that all these outlines intersect the features of the sketch 
already delineated at the proper points. The correct repre- 
sentation of contours is the most difficult part of sketching, 
since these lines are quite imaginary, yet for railroad maps 
they are usually as important as any others. It is desirable 
to know not only the locality of a hill or slope, but also its 
shape, steepness, and height. This information is best 
given by contour lines. A contour is the intersection of the 
surface of the ground by an imaginary level surface. When 
the surface is real, like that of a lake, the intersection is called 
a shore. If the water should rise a certain height a new 
shore would be defined, and rising double that height still 
another shore would result, each of which, on the subsidence 
of the water, would be a contour. A practiced eye is able 
to follow on the ground the course of a contour with all its 
windings; but in sketching them due allowance must be made 
for the foreshortening effect of distance. All contours on 
the same sketch should have the same vertical interval, 
so that by counting them the height of the hill may be known. 
The spaces on the sketch between contours vary as the co- 
tangent of the slope angle, so that the width of the spaces 
indicates the degree of steepness. The contours nearest 
the topographer should generally be sketched first, although 
if there be a shore that is apt to be the best guide to the 
shape of the slopes. If the height of the hill is known and 
the upper contour located, the other contours can be spaced 
between with less difficulty, the proper number being ascer- 
tained by dividing the height by the assumed vertical interval. 
A special line of levels up an inclined ravine or sloping ridge 
to fix the contour points is often of the greatest service in 
obtaining correct results. Other random lines are sometimes 
run to locate the contours more definitely. These should 
be made to cross several contours rather than to trace a 


TOPOGRAPHICAL SKETCHING 303 


single one. Old preliminary lines which have proved useless 
in themselves often furnish by their profiles valuable informa- 
tion in respect to contours. | 

The use of hatchings should be avoided in the sketch- 
book, except to represent precipitous banks, or slight terraces, 
which would not be sufficiently defined by the contour 
system. Hatchings freely used consume too much time, 
and fail to give an accurate idea of either slope or height, 
while they obscure the page for the representation of other 
objects. . 

330. The center line on the page is straight, and for 
sketching purposes the surveyed line on the ground is 
_ assumed to be so also. Slight deflections in the course of a 
preliminary line may be ignored in the sketch; but if a large 
angle occurs it 1s better to terminate the sketch with the 
course, and begin again, leaving a few blank lines between 
the two sketches. On a located line with curves, the sketch 
is continuous. The curved line in the field is represented 
by the straight line on the page, and the radial lines through 
the stations are represented by the parallel lines ruled across 
the page. All objects are sketched at the proper offset 
distance by scale from the center line; but longitudinally 
the sketch is necessarily diminished outside of the curve, 
and magnified inside of the curve. Consequently topo- 
graphical lines which are straight in fact appear curved in 
the sketch, concave to. the center line if inside the curve, 
and convex if outside of it. Such features are correctly 
sketched by means of offsets estimated or measured from 
each station of the curve on the radial lines. This kind 
of distortion creates no confusion if properly done, for in 
making the map, after drawing the curve and the radial - 
lines, the same offsets will give the correct positions of the 
objects delineated. This method is preferable to drawing 
a curved line on the page to represent the center line, as it 
is difficult to draw it correctly; it will cross the ruled lines 
obliquely, rendering them of no service for offsets or scales, 
and moreover is likely to run off the page altogether. 


CHAPTER XIX 
ADJUSTMENT OF INSTRUMENTS 


Every adjustment consists of two processes: first the test, 
and second the correction. .Inasmuch as the amount of cor- 
rection is made by estimation, the test must always be 
repeated until no further lack of adjustment is observable. 


THE TRANSIT 


331. The level tubes should be parallel to the 
vernier plate. 

Test: Place the tubes in position over the leveling screws, 
and turn the latter till the bubbles are centered; turn the 
plate half way around until the tube is parallel to the same 
pair of screws. The bubbles should remain centered; if 
they have retreated— 

Correction: Bring them half way back to the center by 
turning the adjusting screws which attach the tubes to the 
plate. 


The line of collimation. should be perpendicular 
to the horizontal axis. 

Test: Clamp the limb, and by the tangent screws bring 
the intersection of the cross-hairs to cover a well-defined 
point about on a level with the telescope: plunge the tele- 
scope to look in the opposite direction, and note any point 
about on a level with the telescope and about equidistant 
with the first point, which the intersection of the cross- 
hairs now happens to cover. Now unclamp the limb and 
turn the telescope in azimuth, and repeat the above opera- 
tion, using the same first point as before. 

The third point obtained should be identical with the 
second; if not— 

Correction: Move the vertical cross-hair over one 
fourth of the apparent distance from the third to the second 
point, by turning the adjusting screws at the side of the 
telescope. 

304 


ADJUSTMENT OF INSTRUMENTS 305 


_The horizontal axis should be parallel to the ver- 
nier plate. 

Test: After completing the above adjustments level the 
limb, clamp it, and bring the intersection of the cross-hairs 
to cover some high point so that the telescope may be elevated 
to a large angle; depress the telescope and note some point 
on the ground now covered by the intersection of the cross- 
hairs. Now unclamp the limb, turn the telescope in azimuth 
and repeat the above operation, using the same high point 
as before. The third point found should be identical with 
the second; if not— 

Correction: Raise the end of the axis opposite the 
. second point (or lower the other end) by a small amount, 
by turning the adjusting screws in the standard. The amount 
of motion required is determined only by repeated trials 
until the test is satisfied. 

The intersection of the cross-hairs should appear 
in the center of the field of view. 

Test: Bring the cross-hairs into focus and direct the tele- 
scope toward the sky, or hold a sheet of blank paper in front 
of it. If the intersection appear eccentric— 

Correction: Turn the screws (by pairs) which support 

the end of the eyepiece until the desired result is obtained. 


If there be a level on the telescope it should be 
parallel te the line of collimation. 

Drive two stakes equidistant from the instrument in exactly 
opposite directions, and having perfected the previous 
adjustments, level the plate carefully, clamp the telescope in 
about a horizontal position, and observe a rod placed on each 
stake. Have the stakes driven by trial until the rod reads 
alike on both. The heads of the stakes are then on a level. 
Remove the instrument beyond one stake, and set it up in 
line with the two, level the plate, and elevate or depress the 
telescope to a position which will again give equal readings 
on the stakes. The line of collimation.is now level— 

Test: While in this position the bubble of the attached 
level should stand centered; if not— 

Correction: Bring the bubble to the center by turning 
the nuts at one end of the tube, while the cross-hair con- 
tinues to give equal readings, 


306 FIELD ENGINEERING 










THE Y LEVEL 


332. The line of collimation should coincide wi 
the axis of the telescope. 

Test: Clamp the spindle, and bring the intersection | 
the cross-hairs to cover a well-defined point by the tange 
and leveling screws; revolve the telescope half over in t 
Y’s, so that the fayel tube is on top. The intersection of t 
cross-hairs should still cover the point. If either hair h 
departed— 

Correction: . bring it half way back by means of the p: 
of adjusting screws at the extremities of the other hair. 


The attached level should be parallel to the ax 
of the telescope. 

Test: Bring the telescope over one pair of leveling screy 
clamp the spindle, open the clips, and bring the bubble to t 
center. Then gently remove the telescope from the Y, 
and replace it end for end. Ifthe Y’s have not been disturbe 
the bubble should return to the center. If it does not— 

Correction: bring the bubble half way back by turni 
the nuts at one end of the tube. 

But as now the level tube and telescope may only lie 
parallel planes, and yet not be parallel to each other— 

Test: bring the bubble to the center as before, and ti 
the telescope on its axis so as to bring the level tube out 
one side. The bubble zhuuld remain centered. If it | 
departed— 

Correction: bring it back to the center by the adjust: 
screws at the side of the level tube. 


The axis of the telescope should be at rie 
angles to the spindle. 

Test: Having completed the above adjustments (and i} 
before), fasten down the clips, unclamp the spindle, and fii 
the bubble to the center over each pair of leveling scre(s 
in succession, then swing the telescope end for end on : 
spindle. The bubble should settle at the center. If it dis 
not— 

Correction: bring it half way back by the large nutsit 
one end of the bar, 





ADJUSTMENT OF INSTRUMENTS - 307 


THE DUMPY LEVEL 


333. The telescope of the dumpy level cannot be taken 
from the Y’s, consequently there are two adjustments only. 
These are: 


The axis of the bubble should be at right angles 
to the axis of the spindle. 

The test. and correction are the same as for the plate level 
adjustment of the transit. 


The attached level should be parallel to the 
axis of the telescope. 

The test for this adjustment is the same as the correspond- 
ing one for the transit. The correction, however, is made by 
changing’ the position of the reticule which carries the cross- 
hairs, the level vial remaining the same. 


APPENDIX. 


| Verification of eq. (77). 








Eq. (76) i) = Sase =sin @. cosec 4 
sin V 
AG =cos 6. cosec any . sin 6. cot ° - cosec : (764) 
on a =p (cot 6 -\ cot x) (77) 
Verification of eq. (81). ; 
Differentiating eq. (763) 
d2p é 6.2 a Up 
er sin 6 cosec WN cos 6 cot N cosec N + 
0 ‘Ley: 6 
ua sin 6 cot? ° cosec W +73 sin 6 cosec3 WN 
2p 2 +f, (cot? 2 -+coseet &) 
a Pe ar cot @. cot Wty cot wy teosec N 
° d2p 2 Boreas. 2 ) 
e Sexe (— 1-7 cot 6 cot 5; tay (2 cot Nth) 


308 


APPENDIX 309 


Now r= 


dp d2p 
in which substitute for — de’ and for —— qe? , and let 


cot @ bi Wiad G 
N N 


(p?+02(—a)?)3 


2 Cais ee —p2 | —.]| —— = — — 
Pp? F2p?( —a)? —p ( 1 W cot 6 cot Wty (2 cot? < +1) 





Pe (1 + a2)2 
2 02m? 0 1 
1 a? +i cot 6 = cot = WN WN? cot? W aN? 
3 
=f. eee ie ee 
2 48: 1 D 
a (1-+a2)2 
AG Dpediin tea (ios Wea oN, 
"20 aoa (a- W cot x) 
Me prea § CE o2)? 
72 


1 
1—3y2 2 cot 9 


emia fi 0%), ET 


= Bite: wasating oF tte 


=r 


Pia te Re SRR 


weer” 
a 


SS ty 


2 


be Seg ore y 


+ 
















































































































































































































































































































































































































































































































































































i "ss TN foi, | [ue r 
t <I C ] CIOINET NONLIN NC NY NI Ae hal Tal 
INK] TIN Lake \ \) PNiociakent aces | H Ho fai uoial 
2 | | a f f T} 
l 
| | | LN | & 
oN q wo Oy 
— ‘ = —, IN % \r 
| In 7 l 
=tO7 ! : IN | : + o) 4 
fz ihe N Ni PSS NG r} | 
OF NSONS EST SSCS SCN aN " hi} cry im 
Bl T Rel Sh; T r a ct Lt [ | 
+ + . a NIN AL | 1g a 
| : ; J T RED | Won 
- resis SI S x Hy ‘+ 
| el [eat I i | | I | 
| | | [els ~ ~ SN Ng ‘ iN \ {\! Toth 
| coon OSS S SOSN ONCE ETEH 
| | oe | eos Ip J LN D SN N \ \ Wy is GA Th 
| | 22, SS ‘G N\ 1 |! s e nw! ae 
u | T T ) ial 
is 4 ey sans S. N ax \ NIN PACT ri 
T | | 1 a es NY] t ‘ | ] —] - 
raat) T | | | LPs > SW 1 a 
H ae : | ey SSSR 
ay oe 1 = \ a 
| | =i [a CT] SOF } Q ( SNA TULWAVAN mal 
: Bseneyseasaaseres yi SSA SSE : 
LH | ro SSVws RANKS ancuseus 
HH | + 645 NS HH 
; | Gene i Hi 
i =f an Er] . AY 
Ki i Ca | [ >) ; 
| ws SS T 
| Cl | | OSsss We 
ol mim Soeeh CT SSS AA 
PEELE : ooee been Doea Ee 5 Sea seacayeanan en! 
im a hele 7 | | was i NCCE r 
| | SEEuuneueu ene mobs CEESEESSEGESS SSS 
i | let | ae fadtal | | > 
(a [ | [TI | Lt Pe ‘ \ 
| | 
| a 
| t = tear TT 1 -. 
| ia i ] | 7 Q 
| iS | [ SS Ba I rad 
TI tH fail | a fF ihre! 
| IE | | | 

































































































































































"Spf °nQ Ul SUISIig Jojnbuvisy Jo aulnjof 

































































11 


6 ¥ 8 9 
Values of 6 


4) 





PLATE I. 


TRIANGULAR PRISMS 














Yon 































































































































































































































































































Yolume of Triangular Prisms in Cu. Yds, 








































































































































































































































































































H a5 GEnes 20URSZ 7 
A_| fia | aig 
LO -— as 0; 
E PEELE EEE 
4 TV ent | = Aca i 
60+ 7 50 
| ei | ia LF 
fel amr di 25 a TI 
bs ian ALL —| 
{| > J EBZ Cm | i=) cra oO. s u ‘co. Une 
LA = wl le a Go Alert les ON atte 
F) | — 50 
CI 88 B! f =f 
Be (1 
i ae oo ai ag 
iz Eee: ee ae ql ier a a ial chefs 
a : | | Appt 
i i cl a 
| | T |_| ys g} a! 
ary 3 SP VeNOSN ES lai 
230) ot Prt 0 = 
| [ g i 1 a |e wt 
> oe Ct a ane ae== oe s 
SeLeEpe BESEEE Sera Cc Tt 
aT oa Bie! i fons axa | ia is 
20- eat u Via ea \ 20 a 
4 EEE a =H BEE Eee 
I eee EEE EE Peet SERaRe = 
sea Jenses smee= noon SERESEEGEESETTGneeeHeaEEenL. - 
7 TEE EEE BEE Lee eee ee a 
t a ea] J — 
= Te +t a Coo 
EEE EEE Eee) 4 a 











14 15 16 17 18 


193320221 
Values of 6 


225° 23) $24 5725 


26 27 








+ 





aa 4 * 
Bgtte HID 32: eet 


op ee 








boks 
en dime 
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rae an: 
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} As et Caer 


ae Se ae 









ead SO i Soha) ee ate + : 
By i. he LrMy + 


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Mat rn Mien hand 















bese = Te oes 2 eee 2) 7 ay phe Omg SEs 1B ; 


age Hyd fe aratsice ceagttaensiers Basgayest | ee 






















































































































































































































































































































































































































































































































































































































































































i. “1 TT ep TENET S 
a “NVE: aw aia rs COTT-Ors nN eS es 
4 i. Loot 2 les 
Bod mi apg Sak PS 
: ef CNN H OtS : es 
7-g—s—d* 
<a cs Eerie el 
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t | 1 | Oo ro = 
oS Lii= 
LE core oS 
| ea a | Lh, od 
oy | | Tout Ko. | 
co igs , o 
0¢ wre ro) 
t N | | T te jad 
LS re ao 
6Z oS 
} A 
Ly NN A oO 
if | i 
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NS 
Nt f | x aa if 
} = e} 
PSH Les Ss aan a al 
cH CN im u 2 imva 
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1 Fe ia ag CRAR DN | 
= aa Pie 2 [ 
re = Oo S| Pes | LO 
a mz) a K4 N 
ay i s TOS 
rom | i james LO eo ij 
Pere eid EK) eS | NC 
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a | a] 6 f a haa (== Say Sl Ta | il “ 
a Fat | T | | fap) PS I 
UV) | ie Sane os iis es 
cto N €z a 
: CI | ener] = —) 
© el a Tl age , 
N | col : | ad = 22 N 
Bi o o Lay = | | Pe 
oO | ate 7 ee 4 NN ae: 
qeereanati BH PEPE SHINN Ni 
| i Het [ cs 
"Spang us syz6ua7 34 0g sos SUINIOA eee. cot Chee wees 


=D 


Distances Between Slope Stakes 





PLATE Il. -THREE LEVEL SECTIONS 


































































































































































































































































































































































































































































































































































































220-440 f 
A etete C 
; 210-430 
{ ; Hf p @: Z fa | fe 
200-420 . é 8 . 
seta 
190-4102. 444 
oa oe : ai 
[ et 1 HAD ie | 
170-390 VAG : AL 
| Li Al | 
7 | i “ih 
4160-380: ay anyaeaee 
et ARAL ae CHEMO 
1370 e PAPE 
i /\ ile | 
a tape Ct 
A : 
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| AY is ay A le 
AS A 330 
5 Zz H 
~ : 20 
[2 | 
5 [ Dz / [ Bi) aw 
‘Saaecand 310 
1 
TO: » co 
© 29 300 
Anse + AS OAS By { 
ao ap 250 : Poy 
arate Fee 
iz / 4 5 at 
280 
A CoA 
fo a 1 
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zien 26 : 
a | 
Brae cee ire | rE 
a cdanceaecagncdae,ctacsaat 
aeeraee 1240 
ft | 
sceeedorcatenantatast : 
AEEEHEE-EEE A 3 g EHH | +H 

















440 
430 
420 
410 
400 
390 
380 
370 
360 
350 
340 


Volumes for 50 ft. Lengths in Cu.Yds. 


§ 36 37 38 39 40 41 42 43 44 45 46 47 


Distances Between Slope Stakes=D 




































































































































































Base 16. Slope 1% to 1fft4 SEU naWRY a 
Sab dCEad# cbse scat seat coapi onasooatauuetsfatecat 
bossa fasuitanatecai anaffaitceaiisede 
ead aiid (adel FPel naan neue taeg eetpeeet og 
eet cesed oscil feseasseas dice) 04 c2o72¢ce2 40 he 





























































































































Volumes for 50 ft. Lengths in Cu.Yds. ~ 













































































: 130- é i 9 zZ a, | q S Z 5 7 A 
i AA APTA A EAT Ae 
sfc SES REP, TA a A Osa Ae S288 & i 
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‘la as VAe AA Ry file BP AGB 
7 SRG 029 48/48 ZA aa AT 
4-0 
a eee ap 4041 At t Prete 
{ 1:69 | a I | ! | | { 4 ch: 
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CI 4 “220. / i 





17 


18 














1955205 > 219) 22 23 © 24480552 
Distance between Slope Stakes=D 





6. -2Zimeeu 


‘ 
4 








PLATE Ill. THREE LEVEL SECTIONS 

yaaea sera 2201440) Ae 7a 440 
t CAAA Ht Aa Bap Aes 4 Ay 
scarce a: 404304444 AL 430 













































































































































































































































































































































































































































































































































































ACHAEA HHH ry F AH 
LTA Ta | ia an Va 4 [ 
erry “7717200742047 : TA] 420 
A a at f Bear an'san 4 (60000 460 464m Ber, 
~ AE 90- 440-1 AD A pA A A A EAA 
[dar dup adap aserane AH LAH H 
a | 180 40 0- al ral Te i 400 
1 . Z a5 PE ; V/. Ly A 3O A- AL | 
1a i nisi, { 7 Ume4ava _ | CY 390 
Z ssgans | CAAA CA Za 
160-380-445 BoA arvana 1380 
POrsiO7 HK va zt 340 
Hi ALARA ARM OA 
aA “360 A I | £) |_| ya a2 f |Z_] ; 360 
OVA MAM AL Bosse rate 
50 Davy, x) 350 
if A y AY Sin ey L iz ya ae 
oe Z | Hap 4may, 340 
yaar ane Ves trey 
: SAC : S Ze | 330 
ai AAG ATA 
7 ava A; [ WL = 2.0: 
‘a LVL Gey = ner el : 
papas [ a | shor S 
BL eh ey Le arr a CH S 
fea L ee 110 fe 4 Co Heb Lead Hoot ae et CO es 
I | Ft | | A | I 1 | 00 S 
4 | ) | Fe = Ss 4 i s 
iD ‘4 D : cE] si TI P4wl 290 $2: 
| a ie) lA in ad [ Bi | ie S 
1 j 1 LA te} arae esi) af Z = 280 = {1 |_| aa a 
} : 4 imi ae 270 TW * 
LAL AGL VIS HeLa oe EERE & 
| ra 6 ey (2.60 ae = 
4 [ : L AHH F | if i. +H 2 
scaep eee eCao t fd aseeeeeet 
| | ay fi | aeze co {7 - a a E ea FH | § 
aPaat iecat ab7_ ir cat atte ad aca oat cadna ae eatead fasta 
#1 @ VW ee (SSE alata ei 
405458067 oe im 230 a Seis 
asezaecae! capo tHEHHHHH EEE EEE 
































































































































au r 
30 31 32 "33 34 35 36 37 38 39 40 At 42 43 
Distance between Slope Stakes=D 





my 


r, 


































































































































































































































































































Pe 






























































































































































ilu 


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| 



















































































Spun{x Ig Ul SU0I}I0NI09 


a 


(D-D’) 


Distances Out 


PLATE IV. PRISMOIDAL CORRECTIONS 








2 





iy 








2 























05 




































































Fa 


































































































































































































































































































































































































































































































9 10 11 
Distances Out =(D-D’) 

















13 








Corrections in Cubse Yards 





FIELD ENGINEERING _ 


A HANDBOOK OF THE 


THEORY AND PRACTICE OF RAILWAY SURVEYING, 
LOCATION AND CONSTRUCTION 


BY 


WILLIAM H. SEARLES, C.E. 


Member American Society of Civil Engineers 





7 


VENTEENTH EDITION, REVISED AND ENLARGED 
TOTAL ISSUE FIFTY-THREE THOUSAND 


By 
WILLIAM H. SEARLES, C.E, 


AND 


HOWARD CHAPIN IVES, C.E. 


Professor of Railroad Engineering, in the Worcester 
Polytechnic Institute 


VOLUME II 
TABLES 


NEW YORK 
JOHN WILEY & SONS, Inc. 
Lonpon: CHAPMAN & HALL, Liwtrep 
1915 





Copyright, 1880 » 
BY 
JOHN WILEY & SONS 


Copyright renewed, 1908 
By JOHN WILEY & SONS 


Copyright, 1915 
By WinuiAmM H. SEARLES 
and 
HowaARD CHAPIN IvEs 


Composition and Electrotyping 
By THE ScIENTIFIC PRESS (Robert Drummond & Co.), Brooklyn, N. Y. 


Printing and Binding 
By BRAUNWORTH & Co., Brooklyn, N. Y. 


PREFACE 


Tue tables are now published either separately or with 
the text. 

In this edition sixteen new tables have been added. These 
_are: Tables Nos. 11, 12, 18, 19, 22A, 22B, and 31 to 40 in- 
clusive.. Table 18 replaces two tables in the previous edition. 
Tables 8, 9, 43 and 44 have been either slightly changed or 
extended. é 

Tables 11, 12, 31 and 32 have been computed anew. Tables 
34 to 40 inclusive are from Searles’ “ Railroad Spiral,’’ some- 
what extended and rearranged. 

Acknowledgments are made to the American Railway 
Engineering Association for permission to use material in 
Tables 138, 22A, 22B, and 33; and to W. & L. E. Gurley for 
permission to use the material in Table 19, which is from 
‘ their manual. 

The following from the preface to the first edition applies 
‘to other tables. 

Among the tables, Nos. 2, 3, 7, 26, 29 and 48 are 
original. The adoption of versed sines and external secants 
throughout the work, wherever these would simplify the 
formulas, rendered necessary the preparation of tables of 
these functions. The tables of logarithmic versed sines 
and external secants has been computed from ten-place 
logarithmic tables of sines and tangents, so that the last 
decimal is to be relied on, and no pains have been spared 
to make the table thoroughly accurate. 

Tables Nos. 1, 4, 5, 6, 8, 9, 22 and 30 have been re- 
calculated, enlarged, and some of them carried to more 
decimal places than similar tables heretofore published. 
The intention has been to give one more decimal than usual, 
so that in any combination of figures the result of calcula- 
tion might be reliable to the last figure usually required. 


ili 


IV PREFACE 


The tables which have been compiled and rearranged are 
Nos. 15, 16, 17, 24, 25, 41, 42 and 44. The tables of log 
sines and tangents here given are the only six-place tables 
which give the differences correctly for seconds. The table 
of logarithms of numbers is accompanied by a complete 
table of proportional parts, which greatly facilitates inter- 
polation for the fifth and sixth figures. 

In all the tables, whether new or old, scrupulous care has 
been taken to make the last figure correct, and the greatest 
diligence has been exercised by various checks and compari- 
sons to eliminate every error. It is, therefore, hoped and 
believed that a very high degree of accuracy has been ob- 
tained, and that these tables will be found to stand second 
to none in this respect. 

W.#H.S. 
Boca 
Juuy, 1915, 


TABLES 


/ 


PAGE 

Peace OLsets..anc: Orcinatesae sree aero neiaem reine 1 

II. Corrections for Tangents and Externals............. 9 

III. Tangents and Externals to a One-degree Curve....... 10 

EVee bono. @hords range ctualeA rcs ie selon meena 14 

Wee DineareMetlections wrt... te Pe eee ee oe 19 

Vi. Middle Ordinates to Long Chords:........ts<-0».: 20 
Corrections 107 Subchord Ienethsa. 54.02% sec neo os dee 2S 

Vile Curved: Deflections™ ValvoidvArcs..). a)... 42c64e6 a6 24 
ViihieViddicsOrdinates 1orsvallsinidsn ise) cts. oe cierto te he 
ixca Difference inslevationsOl Walla gees an. ace tieekiercts sae 25 

vee inches ir Decimals ota Foot. set. t arts + ciel sale scoe 26 

Ee VEC tri CR GUL V.ES tas srapeeche Ati ees) are oP ene ys 3 Mo eaetens te hors Parl 
ae ACres LOLS trip elo Ete Wider «amet seen 3 cise eo 28 
NEM VIGLOCIEVR ELCIOMCGA, Varker tc eee cme eects eee 29 
Dole Gradeszand.Grade-Anples.. Won. cutwe dade sa ook ate alee 30 
MV Barometric Hergnigswinibeetaaren ake cen a omit aiisets He 32 
XVI. Coefficient of Correction for Atmospheric Tempezature 34 
XVII. Correction for Earth’s Curvature and Refraction..... 34 
Xa Med UCtion, Ob SeAdia ReCAGINGS. = as pocmeers © cc cies ciety cece 35 
MPxXuVMean Retractions meDechinations «ace... sci. s ce 38 
XX. Lengths of Circular Arcs............ OF 5 Rate Mee ea Rt ee 41 
KOM eVMiniibes im. Decimals.of a. Degree risus cols censecaoe cies 42 
XXII. Frog Angles and Switches; Stub Switch............. 3 
XXIIA. Split Switches; Frogs; Theoretical Leads............ 44 
xX MIBe Split) Switches; Practical Weads: 7. 0. .4.62 oe. ces o oes 45 
XXIII. Squares, Cubes, Roots, and Reciprocals............. 46 
Dex Ver Worarithimss OLAINUIMDETSS. «yas 5 deuut eek. nic a hota te ose 8 63 
XXV. Logarithmie Sines, Cosines, Tangents, and Cotangents. 90 
XXVI. Logarithmic Versed Sines and External Secants....... iss) 
XOXGVAl MneNAtUalssinesiand. COsimes;., cls + ai sterekatonic « cucheiaene elehe § 180 
XXVIII. Natural Tangents and Cotangents.................- 189 
X XIX. Natural Versed Sines, and External Secants ......... 201 
XXX. Cubic Yards per 100 Feet in Level Prismoids......... 224 
XXXI. Triangular Prisms; Cubic Yards per 50 Feet......... 231 
XXXII. Prismoidal Corrections; Cubic Yards per 100 Feet.... 241 
Kou Lies igoramstor len gthsorOpirals.n cle ic sic osiehentele & cele 246 
XXXIV. Elements of the Spiral of Chord-length 100........... 247 
Roo vewspiral WMeflection Amelestieys isis ccist a -cere «+ afalalassle Soe 248 
XXXVI. Spiral Degrees of Curve and Coordinates, xz, y........ 250 
XXX Hee Punctions, ofthe Spiral Angie's, 5.22. .cdese ee see 265 


Vv 


XXXVITI. 
XXXIX., 
XL. 

XLI. 
XE 
XLITI. 
XLIV. 


TABLES 


Spiral Tangents, Long Chord, Coordinates p, g....... 
Selected Curves with Proper Spirals and p, q......... 
Equal Length Curves with Spirals 
Geometrical Propositions es caeeie c as oie canon ares 
Lrigonometrical i ormulase ewer ete: ook. eee eee 
Formulas, Curves and Earthwork 
Useful Numbers and Formulas 
Explanation of Tables 


see) a oe 8 + 6 0 06 SO eee 


oN = cre « is) 6.0 0 ¢ ee 6s 6 & 6 


Stoke ‘sl siiel wo 084 © © 6 © ine © O orgie: 


Co: wig 0m, #8 ee ees he (oa elles @€ 8.056 O00 eae 























TABLE I.—RADII, LOGARITHMS, OFFSETS, ETC. 1 
: Loga- | Tan. | Mid. ‘ Loga- | Tang. | Mid. 
Deg. |Radius.| rithm. | Off. | Ord. || Des | Radius.) rithm. | off. | Ord. 
D. R. log. R. t. m. D. R. log. R. t. m. 
O° -0’ | Infinite | Infinite | .000} .000 | 1° 0’) 5729.65 | 3.758128 873 | 218 
1 | 348775. | 5.536274 | .015 | .004 || 1 | 5685.72 750950 | .887 | .222 
2 | 171887. | 5.235244 | .029 | .007 2 | 5544.83 743888 .902 | .225 
3 | 114592. | 5.059153 | .044 | .011 3 | 5456.82 | .736939 | .916 | .229 
4 | 85943.7 | 4.934214 | .058 | .015 4 | 5371.56 . 730100 9381 | .238 
5 | 6874.9 | .837304 | .073 | .018 5 | 5288.92 | .723367 | .945 | .236 
6 | 57295.8 | * .758123 | .087 | .022 6 | 5208.79 (16737 | .960 | .240 
7 | 49110.7 .691176 | .102 | .025 7 | 5181.05 | .710206 974 | .244 
8 | 42971.8 .638184 | .116 | .029 8 | 5055.59 703772 -989 | .247 
9 38197. 2 .582031 | .131 | .033 9 | 4982.33 697432 | 1.004 | .251 
10 | 34877.5 | 4.536274 | .145 | .086 10 | 4911.15 | 3.691183 | 1.018 | .255 
11 | 31252.3 | 4.494881 | .160 | .040 11 | 4841.98 | 3.685023 | 1.033 | .258 
12 | 28647.8 -457093 | .175 | .044 12 | 4774.74 |. .678949 | 1.047 | .262 
13 | 26444.2 422331 | .189 | .047 13 | 4709.33 .672959 | 1.062 | .265 
14 | 24555.4 .890146 | .204 | .051 14 | 4645.69 | .667051 | 1.076 | .269 
.15 | 22918.3 -3860183 | .218 | .055 15 | 4583.7 .661221 | 1.091 | .273 
16 | 21485.9 .332154 | .233 | .058 16 | 4523.44 | .655469 | 1.105 | .276 
17 | 20222.1 .805825 | .247 | 2062 17 | 4464.70 | .649792 | 1.120 | .280 
18 | 19098.6 | ..281002 | .262 | .065 18 | 4407.46 .644189 | 1.184 | .284 
19 | 18093. 4 .257521 | .276 | .069 19 | 4351.67 .638656 | 1.149 | .287 
20 | 17188.8 | 4.235244 | .291 | .073 20 | 4297.28 | 3.633194 | 1.164 | .291 
21 | 16870.2 | 4.214055 | .305 | .076 21 | 4244.23 | 3.627799 | 1.178 | .295 
22 | 15626.1 | .193852 | .3820 | .0380 22 | 4192.47 | .622470 | 1.193 | .298 
23 | 14946.7 .174547 | .835 | .084 23 | 4141.96 | .617206 | 1.207 | .302 
24 | 14324:0-| .156064 | .349 | .087 24 | 4092.66 | .612005 | 1.222 | .305 
25 | 13751.0 | .138335 | .864 | .091 25 | 4044.51 .606866 | 1.286 | .309 
26 | 13222.1 .121302 | .878 | .095 26 | 3997.49 -601787 | 1.251 | .313 
27 | 12732.4 .104911 | .3893 | .098 27 | 3951.54 | .596766 | 1.265 | .316 
28 | 12277.7 .089117 | .407 | .102 28 | 3906.64 .591803 | 1.280 | .320 
29 | 11854.3 | .073877 | .422 | .105 29 | 3862.74 | 586896 | 1.294 | .324 
30 | 11459.2 | 4.059154 | .486 | .109 30 | 3819.83 | 3.582044 | 1.309 | .327 
31 | 11089.6 | 4.044914 | .451 | .113 31 | 8777.85 | 3.577245 | 1.324 | .331 
“32 | 10743.0 .031125 | .465 | .116 32 | 8786.79 | 572499 | 1.338 | .335 
33 | 10417.5 .017762 | .480 | .120 33 | 3696.61 .567804 | 1.3853 | .338 
34 mile 1 | 4.004797 | .495 | .124 34 | 3657.29 | .563160 | 1.3867 | .342 
35 | 9822.18 | 3:992208 | .509 | .127 35 | 3618.80 { .558564 | 1.382 | .345 
36 9519. 34 .979973 | .524 | .131 36 | 3581.10 | .554017 | 1.3896 | .349 
37 | 9291.25 .968074 | .5388 | .185 37 | 3544.19 .549517 | 1.411 | .353 
38 | 9046.75 | .956493 | .553 | .138 88 | 3508.02 1 1545063 | 1.425 | 856 
39 | 8814.78 | .945212 | .567 | .142 39 | 8472.59 | .540654 | 1.440 | .360 
40 | 8594.42 | 3.934216 | .582 |} .145 49 | 3437.87 | 3.536289 | 1.454 | .364 
41 | 8384.80 | 3.923493 | .596 | .149 41 | 3403.83 | 3. petro 1.469 | .367 
42 | 8185.16 | .913027 | .611 | .153 42 | 3370.46 .527690 | 1.483 | .371 
43 | 7994.81 .902808 | .625 | .156 43 | 3337.74 | “BOS153 1.498 | .375 
44 | 7813.11 .892824 | .640 | .160 44 | 3305.65 519257 | 1.518 | .378 
45 | 7639.49 .883065 | .654 | .164 45 | 3274.17 2515101 1'.527 | 2382 
46 | 7473.42 .873519 | .669 | .167 46 | 3243.29 .510985 | 1.542 | .385 
47 | 7314.41 .864179 | .684 | .171 47 | 3212.98 -506908 | 1.556 | .389 
48 | 7162.03 .855036 | .698 | .174 48 | 3183.23 .502868 | 1.571 | .393 
49 | 7015.87 .846082 | .713 | .178 49 | 3154.03 .498866 | 1.585 | .396 
50 | 6875.55 | 3.837308 | .727 | .182 50 | 3125.36 | 3.494900 | 1.600 | .400 
51 | 6740.74-| 3.828708 | .742 | .185 51 | 3097.20 | 3.490970 | 1.614 | .404 
52 | 6611.12 .820275 | .756.| .189 52 | 3069.55 -487075 | 1.629 | .407 
53 | 6486.38 .812002 | .771 | .193 53 | 3042.39 .483215 | 1.643 | .411 
54 | 6366.26 .803885 | .785 | .196 54 | 3015.71 .479389 | 1.658 | .414 
| 55 | 6250.51 .795916 | .800 | .200 55 | 2989.48 .475596 | 1.673 | .418 
56 | 6138.90 788091 | .814 | .204 56 | 2963.72 .471886 | 1.687 | .422 
57 | 6031.20 . 780404 | .829 | .207 57 | 2938.39 .468109 | 1.702 | .425 
58 | 5927.22 772851 | .844 | .211 58 | 2913.49 | .464413 | 1.716 | .429 
59 | 5826.76 | .765427 | 1858 | .215 59 | 2889.01 .460749 | 1.731 | .433 
60 | 5729.65 | 3.758128 | .873 | .218 60 | 2864.98 | 3.457115 |.1.745 | .436 
I 
























































2 


Deg. 
D. 





200" 


bea 
SCUEMHIOMTR WOH 











TABLE I.—RADII, LOGARITHMS, OFFSETS, ETC. 


Radius. 


R. 


2864.93 


2841 .26 
2817.97 
2795.06 
2772.53 
2750.35 
2728 52 

707 04 
2685.89 
2665.08 
2644.58 
2624.39 
2604.51 
2584.93 


1910.08 








Loga- | Tang. | Mid. 


rithm. Off. | Ord. 
log. R. t. m. 
8 457115 | 1.745 | .486 
.453511 | 1.760 | .440 
.449987 | 1.774 | .444 
.446392 | 1,789 | .447 
.442876 | 1.803 | .451 
.439388 | 1.818 | .454 
.435928 | 1.832 | .458 
.482495 | 1.847 | .462 
.429089 | 1.862 | .465 
.425710 | 1.876 | .469 
8.422356 | 1.891 | .473 
3.419029 | 1.905 | .476 
-415727 | 1.920 | .480 
.412449 | 1.9384 | .484 
.409197 | 1.949 | .487 
.405968 | 1.963 | .491 
.402763 | 1.978 | .494 
.899582 | 1.992 | .498 
1896424 | 2.007 | .502 
.893289 | 2.022 | .505 
3 390176 | 2.036 | .509 
3.387085 | 2.051 | .513 
.3884016 | 2.065 | .516 
. 880969 | 2.080 | .520 
.38779438 | 2.094 | .524 
.8749388 | 2.109 | .527 
.871954 | 2.128 | .531 
.868990 | 2.1388 | .534 
.866046 | 2.152 | .5388 
.863122 | 2.167 | .542 
8.360217 | 2.181 | .545 
3.357382 | 2.196 | .549 
.804466 || 2.211 | .5538 
.801618 | 2.225 | .556 
.848789 | 2.240 | .560 
.845979 | 2.254 | .564 
.8438187 | 2.269 | .567 
.840412 | 2.288 | .571 
.887655 | 2.298 | .57 
.884916 | 2.312 | .578 
3.3821938_| 2.3827 | .582 
8.329488 | 2.341 | .585 
.826799 | 2.356 | .589 
.024127 | 2 871 | .593 
.821471 | 2.885 | .596 
.8188382 | 2.400 | .600 
.816208 | 2.414 | .604 
.813600 | 2.429 | .607 
.3811008 | 2.443 | .611 
.808431 | 2.458 | .614 
3.305869 | 2.472 | .618 
3.803823 | 2 487 | .622 
.800791 | 2.501 | 625 
298274 | 2.516 | .629 
.295771 | 2.580 | .6383 
.293283 | 2.545 | .636 
.290809 | 2.560 | .640 
. 2883849 | 2.57 644 
.285902 | 2.589 | .647 
.283470 | 2.603 | .651 
8.281051 | 2.618 | .654 














ioe) 
° 
= 








a 
(on lole sh Forks) a eer i 




















Radius. 


R. 


1910.08 
1899.53 
1889.09 
1878.77 
1868 .56 
1858.47 
1848.48 
1838 .59 
1828.82 
1819.14 
1809 .57 


1800.10 
1790.73 


1719.12 


1710.57 
1702.10 
1693.72 
1685 .42 
1677.20 
1669.06 
1661.00 
1653.01 
1645.11 
1637.28 


1629.52 
1621.84 
1614.22 
1606.68 
1599.21 
1591.81 
1584.48 
1577.21 
1570.01 
1562.88 


1555.81 
1548.80 
1541 .86 
1534.98 
1528.16 
1521.40 
1514.70 
1508.06 
1501.48 
1494.95 


1488.48 
1482.07 
1475.71 
1469.41 
1463.16 
1456.96 
1450.81 
1444.72 
1438.68 
1482.69 








Loga- 
rithm. 


log. R. 


8.281051 
278646 
276253 
273874 
.271508 
269155 
. 266814 
-264486 
262170 
259867 

8.257576 


3.255296 
258029 


287481 
8.235305 


3.233140 
230985 
228841 
226707 
224584 
222472 
220369 
218277 
216195 

3.214122 


3.212060 
210007 
207964 
205930 
203906 
201892 
199886 
197890 
195903 

8.193925 


3.191956 
189996 
- 188045 
. 186108 
184169 
182244 
180327 
178419 
176519 
174627 


wo w 
~t 
() 
~3 
is 


:157963 
3.156151 


Tang. | Mid. 
Off. | Ord 
t. m. 
2.618 | 654 
2.6382 | 658 
2.647 | 662 
2.661 | 665 
2.67 .669 
2.690 | .673 
2.705 | .676 
2.719 | .680 
2.734 | .684 
2.749 | .687 
2.7638 | .691 
2.707 .694 
2.792 | .698 
2.807 | .702 
2 821 | 705 
2.886 | .709 
2.850 | .713 
2.865 | .716 
2.879 | .'720 
2.894 | .723 
2.908 | .727 
2.923 | .731 
2/938 |: 7734 4 
2.9528 | .788 
2.967 | .742 
2.981 | .745 
2.996 | .749 
8.010 | .753 
3.025 | 2756 
3.039 | .760 
3.054 | ,763 
3.068 | .767 
3.083 | eC 
3.097 | .774_ 
8.112 |} .778 
82127 | 782 
8.141 | .785 
3.156 | .789 
3.170 | .793 
8.185} .796 
8.199 | .800 
3.214 | .803 
8.228 | .807 
3.243 | .811 
3.257 | .814 
8.272 | .818 
3.286 | .822 
3,301 | 825 
3.316 | 824 
3:330 | (832 
3.345 | .836 
3.359 | .840 
3.074 | .843 4 
3.388 | .847 
3.403 | 851 
8.417 | .854 © 
8.482 | .858 
3.446 | .862 — 
3.461 | .865 
8.475 | .869 
3.490 | -.872 








TABLE I.—-RADII, LOGARITHMS, OFFSETS, ETC. 





Deg. 
R. 


4° 
1426.74 
1420.85 
1415.01 
1409.21 
1463.46 
1397.76 
1392.10 
1386 .49 
1380 .92 
1375.40 


1369 .92 
1364.49 
1359.10 
1353.75 
1348.45 
1343.18 
1337 96 
1332.77 
1327.63 
1322.53 


1317.46 
1312.43 
1307.45 
1302.50 
1297.58 
1292.71 
1287.87 
1283.07 
1278.30 
1273.57 


1268.87 
1264.21 
1259.58 
1254.98 
1250.42 
1245.89 
1241.40 
1236.94 


wer 
SCOMVIROTMPWAOHO 


1228.11 
1223.74 
1219, 40 
1215.09 
1210.82 
1206.57 
1202.36 
1198.17 
1194.01 
1189.88 
1185.78 


1181.71 
1177 66 
1173.65 
1169.66 
1165.70 
1161 76 
1157.85 
1153.97 
1150.11 
1146.28 


Radius. 


1482.69 


1232.51 | 


Loga- 
rithm, 


log. R. 


‘3.136697 


3.121404 


(3.119738 
118078 
116424 
LATIZ 
118136 
111501 
. 109872 
. 108249 
. 106632 

3.105022 


3.108417 
101818 
100225 
-098638 
097057 
.095481 
093912 
092347 
.090789 

5.089286 


3.087689 
.086147 
084610 
083079 
.081553 
.080033 
078518 
077008 
075504 

3.074005 


3.072511 
.071022 
069538 
.068059 
066585 
065116 
063653 
062194 

060740 

3.059290 





3.156151 



































Tang.| Mid. | : | Loga- Tang. | Mid. 
Off. | Ord. | Deg. | Redtus: Wnenmae) Of: | Ord, 
t. m. D. R. log, R. t. m. 
3.490 872 |) he O/| 1146.28 )8.059290 | 4.362 | 1 091 
3.505 | 876 1 | 1142.47 | .057846 | 4.376 | 1.094 
3.519 .880 2 | 1188.69 | .056407 | 4.391 | 1.098 
3.534 | .883 3 | 1184.94 | .054972 | 4.405 | 1.102 
3.548 | .887 4 | 1131.21 | 053542 | 4.420 | 1.105 
3.563 | .891 5 | 1127.50 | .052116 | 4.485 | 1.109 
3.57 894 6 | 1128.82 | .050696 | 4.449 | 1.112 
8.592 .898 7 | 1120.16 | .049280 | 4.464 | 1.116 
3.606 902 8 | 1116.52 | .047868 | 4.478 | 1.120 
3.621 905 9 | 1112.91 | .046462 | 4.493 | 1.123 
3.635 .909 10 | 1109.33 {3.045059 | 4.507 | 1.127 
3.650 | .912 11 | 1105.76 (3.048662 | 4.522 | 1.181 
3 664 .916 12 ; 1102.22 | .042268 | 4.536 | 1.134 
3.679 920 13 | 1098.70 |. .040880 | 4.551 | 1.138 
3.693 | .928 14 | 1095.20 | .039495 | 4.565 | 1.142 
3.708 2 15 | 1091.73 | .038115 | 4.580 | 1.146 
3.723 | .981 16 | 1088.28 | .036740 | 4.594 | 1.149 
3.7386 | .984 17 | 1084.85 | .035368 | 4.609 | 1.153 
3.752 | 938 18 | 1081.44 | .034002 | 4.623 | 1 157 
3.766 942 19 | 1078.05 | .082639 | 4.638 | 1.160 
3.781 945 20 | 1074.68 |3.031281 | 4.653 | 1.164 
3.795 949 21 | 1071.34 [3.029927 | 4.667 | 1.168 
3.810 952. 22 | 1068.01 | .028577 | 4.682 | 1.171 
3.824 | 956 23 | 1064.71 | 027231 | 4.696 | 1.175 
3.839 | .960 24 | 1061.43 | .025890 | 4.711 | 1.179 
3.853 | .963 25 | 1058.16 | .024552 | 4.725 | 1.182 
3.868 967 26 | 1054.92 | -.028219 | 4.740 | 1.186 
3.882 | .971 27 | 1051.70 | .021890 | 4.754 | 1.190 
3.897 974 28 | 1048.49 | .020565 | 4.769 | 1.193 
3.911 978 29 | 1045.31 | .019244 | 4.783 | 1.197 
3.926 982 30 | 1042.14 |8.017927 | 4.798 | 1.200 
3.941 985, 31 | 1039.00 {3.016614 | 4.812 | 1.204 
3.955 | .989 32 | 1035.87 | .015305 | 4.827 | 1.208 
3.970 | .993 33 | 1082.76 | .013999 | 4.841 | 1.211 
3.984 996 34 | 1029.67 | .012698 | 4.856 | 1.215 
3.999 | 1.000 35 | 1026.60 | .011401 | 4.870 | 1.218 
4.013 | 1.003 36 | 1023.55 | .010107 | 4.885 | 1.222 
4.028 | 1.007 37 | 1020.51 | .008818 | 4.900 | 1.226 
4.042 | 1.011 88 | 1017.49 | .0075382 | 4.914 | 1.229 
4.057 | 1.014 39 | 1014.50 | .006250 | 4.929 | 1.283 
4.071 | 1.018 40 | 1011.51 |8.004972 | 4.943 | 1.287 
4.086 | 1.022 41 | 1008.55 |3.003698 | 4.958 | 1.240 
4.100 |} 1.025 42 | 1005.60 | .002427 | 4.972 | 1.244 
4.115 | 1.029 43 | 1002.67 |8.001160 | 4.987 | 1.247 
4.129 | 1.082 44 | 999.762 |2.999897 | 5.001 | 1.251 
4.144 | 1.036 45 | 996.867 | .998637 | 5.016 | 1.255 
4.159 | 1.040 46 | 993.988 | .997381 | 5.0380 | 1.258 
4.173 | 1.048 47 | 991.126 | .996129 | 5.045 | 1.262 
4.188 | 1.047 48 | 988.280 | .994880 | 5.059 | 1.266 
4.202 | 1.051 49 | 985.451 | .998635 | 5.074 | 1 269 
4.217 | 1.054 50 | 982.688 |2.992393.| 5.088 | 1.278 
4.231 | 1.058 51 | 979.840 |2.991155 | 5.103 | 1.277 
4.246 | 1.062 52 | 977.060 | .989921 | 5.117 | 1.280 
4.260 | 1.065 53 | 974.294 | .988690 | 5.182 | 1.284 
4.275 | 1.069 54 | 971.544 | .987463 | 5.146 | 1.288 
4.289 | 1.073 55 | 968.810 | .986288 | 5.161 | 1.291 
4.304 | 1.076 56 | 966.091 | .985018 | 5.175 | 1.295 
4.318 | 1.080 57 | 963.887 | .988801 | 5.190 | 1.298 
4.333 | 1.083 58 | 960.698 982587 5.205 | 1.302 
4 347 | 1.088 59 | 958.025 | .981877 | 5.219 | 1.306 
4.362 | 1.091 60 | 955.366 (2.980170 | 5.234 | 1.309 

















4 


TABLE I.—RADII, LOGARITHMS, OFFSETS, ETC. 








60 





Radius. 
R. 


955 .866 
952.722 
950.093 
947.478 
944.877 
942.291 
939.719 
937.161 
934.616 
932.086 
929.569 


927.066 
924.576 
922.100 
919.637 
917.187 
914.750 
912.326 
909.915 
907.517 
905.131 


902.758 
900.397 
898.048 
895.712 
893.388 
891.076 
888.776 
886.488 
884.211 
881.946 


879.693 
877.451 
875.221 
873.002 
870.795 
868 .598 
866.412 
864.238 
862.075 
859. 922 


857.780 
855.648 
853.527 
851.417 
849.317 
847.228 
845.148 
843.080 
841.021 
838 . 972 


836.933 
834.904 
832.885 
830.876 
828 .C76 
826.886 
824.905 
822.934 
820.973 
819.020 


Loga- 
rithm. 


log. R. 


2.980170 


978966 
977706 
976569 
975375 
974185 
972998 
971814 
9706383 
. 969456 
2.968282 


2.967111 
(965943 
964778 
963616 
"962458 
961303 
960150 
959001 
‘957855 
‘956711 
2.955571 
954434 
953300 
952168 
"951040 
949915 
948792 
947673 
946556 
2.945442 


2.944331 
948223 
.942118 
941015 
.939916 
. 938819 
937725 
. 936633 
935545 


cS) 


2.934459 


2.933376 
. 932295 
931218 
. 930142 
929070 
928000 
. 926933 
925869 
. 924807 

2.923747 


2.922691 
. 921637 
920585 
919536 
918489 
917446 
916404 
.915865 
914829 





2.913295 





Tang. 
Off. 


t. 








feel fre fare fred feck Pad Pek fad fed Pe PL 
co 
rw) 
~ 


Peet RR Re fe PP ERR 
oo Oe 
ioe) 
ri) 


D. R. 





1 | 817.077 
2 | 815.144 
3 | 813.219 
4 | 811.303 
5 | 809.397 
6 | 807.499 
7 | 805.611 
8 | 803.731 
9 | 801.860 
10 | 799.997 
11 | 798.144 
12 | 796.299 
13 | 794.462 
14 | 792.634 
15 | 790.814 
16 | 789.003 
17 | 787.200 
18 | 785.405 
19 783.618 
20 | 781.840 
21 | 780.069 











31 | 762.797 
32 | 761.112 
33 | 759.434 
34 | 757.764 
35 | 756.101 
36 | 754.445 
37 | 752.796 
38 | 751.155 
39 | 749.521 
40 | 747.894 


41 | 746.274 
42 | 744.661 
43 | 748.055 
44 | 741.456 
45 | 739.864 
46 | 738.279 
47 | 736.701 
48 | 735.129 
49 | 733.564 
50 | 732.005 


51 | 780.454 
52 | 728.909 
53 | 727.3870 
54 | 725.838 
55 | 7247312 
56 | 722.7938 
57 | 721.280 
58 | 719. ae 
59 | 718.2 














= 


—— — 


Deg. | Radius. 


7° 0) 819.020 





Loga- 
rithm. 


log. R. 


2.918295 


912263 
911234 
910208 
909183 
908162 
907142 
906125 
905111 
. 904098 
. 903089 


902081 


wo 0 


“894105 
2.893118 


2.892183 
891151 
890171 
889193 
888217 
887244 
886272 
885303 

. 884336 

2.883371 


2.882409 
.881448 
880490 
879584 
878580 
877627 
876678 
875730 
874784 
2.873840 


2.872898 
-871959 
871021 
870086 
869152 
868221 
867291 
.866363 

865438 

2.864514 


2.863593 
862673 
861755 
860840 
859926 
859014 
858104 
857196 

6290 


85 
60 | 716. 6 2.855385 


Tang. 
Off. 


ice 

















ol alan eel al ool eel el oe el a a el a ee 








TABLE Lee WADI; oe ee OFFSETS, ETC. 










































































: Loga- |Tang.| Mid. | ._. | Loga- | Tang, 
Deg. | Radius. rachra, Of Ord. Deg. | Radius. isin OF 
D. R. log. R. t. m. D. R. log. R. t. 
8° 0’| 716.779 |2.855385 | 6.976 | 1.746 || 9° 0/| 687.275 |2.804327 | 7.846 
1 | 715.291 | .854483 | 6.990 | 1.749 1 | 636.099 | .803525 | 7.860 
2 | 713.810 | .853583 | 7.005 | 1.753 2 | 634.928 | .802724 | 7.87 
3 | 712.385 | .852684 | 7.019 | 1.756 3 | 633.761 | .801926 | 7.889 
4 | 710.865 | .851787 | 7.084 | 1.761 4| 682.599 | .801128 | 7.904 
5 | 709.402 | .850892 | 7.048 | 1.764 5 | 631.440 | .800332 | 7.918 
6 | 707.945 + .849999 | 7.063 | 1.768 6 | 630.286 | .799588 | 7.933 
7% | 706.493 | .849108 | 7.077 | 1.771 7 | 629.186 | .798745 | 7.947 
8 | 705.048 | .848219 | 7.092 | 1.77 8 | 627.991 | .797953 | 7.962 
9 | 703.609 | .847331 | 7.106 | 1.77 9 | 626.849 | .797163 | 7.976 
10 | 702.175 |2.846445 | 7.121 | 1.782 10 | 625.712 |2.796374 | 7.991 
11 | 700.748 [2.845562 | 7.135 | 1.786 11 | 624.579 |2.'795587 | 8.005 
12 | 699.3826 | .844679 | 7.150 | 1.790 12 | 623.450 | .794801 | 8.020 
13 | 697.910 | .8438799 | 7.164 | 1.793 13 | 622.325 | .794017 | 8.034 
14 | 696.499 | .842921 | 7.179 | 1.797 14 | 621.203 | .793234 | 8.049 
, 15 | 695.095 | .842044 | 7.193 | 1.801 15 | 620.087 | .792453 | 8.063 
16 | 693.696 | .841169 | 7.208 | 1.804 16 | 618.974 | .791673 | 8.078 
17 | 692.302 | .840296 | 7.222 |.1.807 17 | 617.865 | .790894 | 8.092 
18 | 690.914 |. .839424 | 7.237 | 1.811 18 | 616.760 | .790117 | 8.107 
19 | 689.532 | .838555 | 7.251 | 1.815 19 | 615.660 | .789341 | 8.121 
20 | 688.156 |2.837687 | 7.266 | 1.819 20 | 614.563 |2.788566 | 8.136 
21 | 686.785 |2.836821 | 7.280 | 1.822 21 | 618.470 2.787793 | 8.150 
22 | 685.419 | .835956 | 7.295 | 1.826 22 | 612.3880 | .787021 | 8.165 
23 | 684.059 | .835093 | 7.309 | 1.829 23 | 611.295 | .786251 | 8.179 
24 | 682.704 | .834232 | 7.324 | 1.833 24 | 610.214 | .785482 | 8.194 
25 | 681.354 | .833373 | 7.3838 | 1.837 25 | 609.136 | .784714 | 8.208 
26 | 680.010 | .832515 | 7.353 | 1.840 26 | 608.062 | .783948 | 8.223 
7 | 678.671 | .831660 | 7.3867 | 1.844 27 | 606.992 | .783183 | 8.237 
28 | 677.338 | .680805 | 7.382 | 1.848 28 | 605.926 | .782420 | 8.252 
29 | 676.008 | .829953 | 7.396 | 1.851 29 | 604.864 | .781657 | 8.266 
80 | 674.686 |2.829102 | 7.411 | 1.855 30 | 603.805 |2.780897 | 8.281 
31 | 673.369 !2.828253 | 7.425 | 1.858 31 | 602.750 |2.780187 | 8.295 
© 82 | 672.056 | .827405 | 7.440 | 1.862 32 | 601.698 779379 8.310 
33 | 670.748 | .826560 | 7.454 | 1.866 33 | 600.651 | .778622 | 8.324 
34 | 669.446 | .825715 | 7.469 | 1.869 | 34 | 599.607 7867 8.339 
35 | 668.148 | .824873 | 7.483 | 1.87 35 | 598.567 | .777112 | 8.353 
36 | 666.856 | .824082 | 7.508 | 1.877 36 | 597.530 | . 1716360 8.368 
37 | 665.568 | .823193 | 7.512 | 1.880 37 | 596.497 | .'775608 | 8.382 
88 | 664.286 | .822355 | 7.527 | 1.884 88 | 595.467 | .774858 | 8.397 
39 | 663.008 | .821519 | 7.541 | 1.887 389 | 594.441 | .774109 | 8.411 
40 | 661.736 |2.820685 | 7.556 | 1.892 40 | 593.419 [2.773361 | 8.426 
41 | 660.468 |2.819852 | 7.570 | 1.895 41 | 592.400 2.772615 | 8.440 
42 | 659.205 | .819021 | 7.585 | 1.899 42 | 591.384 | .771870 | 8.455 
43 | 657.947 | .818191 | 7.599 | 1.908 43 | 590.372 | .771126 | 8.469 
44 | 656.694 | .817363 | 7.614 | 1.906 44 | 589.364 | .770883 | 8.484 
45 | 655.446 | .816537 | 7.628 | 1.910 45 | 588.359 | .769642 | 8.498 
46 | 654.202 | .815712 | 7.643 | 1.914 46 | 587.357 | .768902 | 8.513 
47 | 652.963 | .814889 | 7.657 | 1.918 47 | 586.359 | .768164 | 8.527 
48 | 651.729 | .814067 | 7.672 | 1.921 48 | 585.364 | .767426 | 8.542 
49 | 650.499 | .813247 | 7.686 | 1.924 49 | 584.373 | .766690 | 8.556 
50 | 649.274 2.812428 | 7.701 | 1.928 50 | 583.385 |2.765955 | 8.57 
51 | 648.054 |2.811611 | 7.715 | 1.982 51 | 582.400 |2.'765221 | 8.585 
52 | 646.838 | .810796 | 7.730 | 1.935 52 | 581.419 | .764489 | 8.600 
53 | 645.627 | .809982 | 7.744 | 1.939 53 | 580.441 | .768758 | 8.614 
54 | 644.420 | .809169 | 7.759 | 1.943 54 | 579.466 | .763028 | 8.629 
55 | 643.218 | .808358 | 7.773 | 1.946 55 | 578.494 | .762299 | 8.643 
56 | 642.021 | .807549 | 7.788 | 1.950 56 | 577.526 | .'761572 | 8.658 
57 | 640.828 | .806741 | 7.802 | 1.953 57 | 576.561 | .760845 | 8.67 
58 | 639.639 | .805935 | 7.817 | 1.957 58 | 575.599 | .760120 | 8.687 
59 | 638.455 | .805130 | 7.831 | 1.961 59 | 574.641 | .759397 | 8.701 
60 | 637.275 |2.804327 | 7.846 | 1.965 60 | 573.686 |2.758674 | 8.716 




















6 


TABLE I.—RADII, LOGARITHMS, OFFSETS, HTC. 











F Loga- 

Deg. | Radius. or 
D. R. log. R. 
10° 0’ | 573.686 [2.758674 








571.784 | 757282 
569.896 | .755796 
568.020 | .754364 
566.156 | .'752937 
564.305 | .751514 
562.466 | .750096 
560.638 | .748683 
558.823 | .74 7274 
557.019 |2.745870 
558.227 |2.744471 
553.447 | .748076 
551.678 |. .741686 
549.920 | .740300 
548.174 | .738918 
546.438 | .737541 
544.714 | .736169 
543.001 | .734800 
541.298 | .733436 
539.606 | 2.732077 


537.924 |2.730721 
536.253 | .729370 
534.593 | .72802: 
532.943 | .726681 
531.303 | .725342 
529.673 | .724008 
528.053 | .722677 
526.443 | .721351 
524.848 | .720029 
523.252 {2.718711 


521.671 '2.717397 
520.100 | .716087 
518.539 | .714781 
516.986 | .713479 
515.443 | .712181 
513.909 | .710887 
512.385 | 709596 
510.869 | .708310 
509.363 | .707027 
507.865 |2.705748 
506.376 |2.704473 
504.896 | .703202 
503.425 | .701984 
501.962 | 700671 
500.507 | .699410 
499.061 | .698154 
497.624 | .696901 
496.195 | .695652 
494.774 | .694407 
493.361 |2.693165 


491.956 |2.691926 
490.559 | .690692 
489.171 | .689460 
487.790 | .688233 
486.417 | .687008 
485 051 | .685788 
483.694 | .684570 
482.344 | .683357 
481.001 | .682146 
479.666 | .680939 
478.339 12.679735 


co) 





w 2 














Tang. 
Off. 
t. 











Mid. 
Ord. 
m. 





cw) 
ee 
—_ 
lor) 

















478 .339 
477,018 
475.705 
474,400 
473.102 
471.810 
470.526 
469.249 
467.978 
466.715 


465 .459 
464.209 
462° 966 
461.729 
460.500 
459.276 
458.060 
456.850 
455.646 
454.449 


453.259 
452.073 
450.894 
449 722 
448.556 
447 395 
446 241 
445.093 
443.951 
442.814 


441.684 
440 559 
439 .440 
438 .326 
437.219 
436.117 
435 .020 
433 .929 
482 844 
431.764 


430. 690 
429.620 
428.557 
427.498 
426 445 
425.396 
424.354 
423.316 
422.283 
421.256 
420.238 
419 215 
418.203 
417.195 
416.192 
415.194 
414.201 
413.212 
412.229 
411.250 


Loga- 
rithm, 


log. R. 


2.679735 
. 678535 
677338 
.676145 
674954 
673767 
672584 
671403 
- 670226 

2.669052 

2.667881 
.666713 
. 665549 
664387 








- 605229 
662074 
- 660922 
-659773 
658628 
2.657485 


2.656345 
- 655208 
.654075 
652944 

. 651816 
- 650691 
-649570 
648451 
647335 

2.646221 











2.645111 
644004 
642899 
.641798 
. 640699 
- 639603 
. 638510 
637419 
636331 

2.685246 


2.634164 
. 633085 
632008 
. 630934 
.629863 
628794 
627728 
. 626665 
625604 

2.624546 


2.623490 


614106 


410.275 |2.613075 


Tang. 
Off. 








12.158 
12,187 


2.€03 


2.985 
2.992 


s 





eooococeo 
oe 
SEES 


G2 0 G8 G8 Co CO eo 9 oo 
co 
o> 


es 





TABLE I.—-RADII, LOGARITHMS, OFFSETS, ETC. 


7 








; Loga- 
Deg. Radius. nt 
D. R. log, R. 
i 
14° 0’ | 410.275 |2.613075 
2 | 409.306] .612048 
4 | 408.341 | .611023 
6 | 407.380 | .610000 
8 | 406.424 .608980 
10 | 405.473} .607962 
12 | 404.526] .606946 
14 | 403.583] .605933 
16 | 402.645 | 604923 
18 | 401.712) .603914 
20 | 400.782 |2.602908 
22 | 399.857 | .601995 
24 | 398.937 .600904 
26 | 398.020} .599905 
28 | 397.108} .598908 
30 | 396.200} .597914 
82 | 399.296} .596922 
34 394.396 | .595933 
86 | 393.501-| .594945 
38 Gunes .593960 
40 | 391.722 |2.592978 
42 | 390.838 | .591997 
44 389.959| .591019 
46 | 389.084| .590043 
48 388.212} .589069 
50 | 387.345 | .588097 
52 | 386.481 | .587128 
54 | 385.621] .586161 
56 384.765} .585196 
53 | 383.913 | .584233 
15° 0 | 383.065 |2.583272 
2 | 382.220) .582314 
4 | 381.380] .581358 
6 | 380.543 | .580403 
8 | 379.709 | .579451 
10 | 378.880} .578501 
12 378.054 | .577553 
14 | 377.231 | .576608 
16 376.412} .575664 
18 | 375.597 | .574722 
20 374.786 |2.573783 
22 | 373.977 | .572845 
24 | 373.173 | .571910 
26 | 372.372 | .570977 | 1: 
28 371.574} .570045 
30 370.780 | .569116 
32 869.989 | .568189 
34 | 369.202} .567264 
36 | 368.418 
38 | 367.637 | .565419 
40 | 366.859 |2.561500 
42 366.085 | .563582 
44 365.315 | .562667 
46 | 364.547 | .561754 
48 | 363.783 | .560843 
50 | 363,022 | .559933 
52 | 362.264) 559026 
54 | 361.510} .558120 
56 | 360.758) .557216 
58 | 360.010} .556315 
60 hod 265 (2.555415 











"566340 | 13.57 








Tan, | Mid. 
Off. | Ord. 


t. m. 





13.629 3.423 
13.658 | 3.431 
13.687 | 3.438 
13.716 | 3.445 
13.744 | 3.452 
13.773 | 3.460 
13,802 | 3.467 
13.831 | 3.474 
13.860 | 3.482 
13.889 | 3.489 
13.917 | 3.496 




















i? 











Radius. 


R. 


359, 265 


358 523 
897,784 
357,048 
356.315 
355.585 
354. 859 
354.135 
353.414 
352.696 
351.981 
351.269 
350.560 
349.854 
349.150 
348.450 
847.752 
847.057 
346.365 
345.676 
344.990 
344.306 
343.625 
342.947 
8422 Die 

341, 598 
340,928 
340.260 
339.595 
308.933 
338.273 
337.616 
336.962 
336.310 
335.660 
335.013 
334,369 
333.727 
333.088 
382.451 
331.816 
331.184 
330.555 
329.928 
829.303 
828 689 
328.061 
827.443 
826.828 
826.215 


825.604 
324.996 
324.390 
823.786 
323.184 
822.585 
821.989 
821.394 
320.801 
320.211 
319.623 





Loga- 
rithm, 


log. R. 





2.555415 
.554517 
.958621 
002727 
551834 
.550944 
.550055 
-549169 
548284 
547401 


2.546519 
545640 
.544762 
.543887 
.543013 
.542140 
-541270 
.540401 
.539535 
538670 


2.537806 


.536945 
536085 
530227 
584370 
.533516 
532668 
531811 
530962 
.530114 


2.529268 
528424 
.527581 
.526740 
.525900 
525062 
524226 
.523392 
522559 
521728 


2.520898 
-520070 
-519244 
518419 
517596 
516774 
.515954 
515136 
.514319 
.518504 


2.512690 
.511878 
511067 
.510258 
509451 
508645 
.507840 
507037 
506236 
505436 

2.504638 


13.917 
13.946 
13.975 
14.004 


14.033 |" 


14.061 
14.090 
14.119 
14.148 
14.177 


14.205 
14.234 
14.263 
14.292 
14.320 
14.349 
14.378 





14.407 
14.436 
14.464 


14.493 
14.522 
14.551 
14.580 
14.608 
14.637 
14.666 
14.695 
14.723 
14.752 


14.781 
14.810 
14.838 
14.867 
14.896 
14.925 
14.954 
14.982 
15.011 
15.040 


15.069 
15.097 
15.126 
15.155 
15.184 
15.212 
15.241 
15.270 
15.299 
15.3827 


15.356 
15.385 
15.414 
15.442 
15.471 
15.500 
15.529 
15.557 
15.586 
15.615 





15.643 














8 


Deg. |Radius. 


D. R. 





319.037 
318.453 
317.871 
317.292 
316.715 
316.139 
315.566 
314.993 
314.426 


| 813.295 
312.732 
| 312.172 
311.613 
311.056 
310.502 
309.949 
309.399 
308.850 


308 . 303 
307.759 
307.216 
306.675 
306. 136 
305.599 
305 . 064 
304.531 
804.000 
303.470 


302.943 
302.417 
| 801.893 
301.371 
300.851 
300.333 
299.816 
299 . B02 
298 .'789 
298.278 


297.768 
297 .260 
296.755 
| 296.250 
295.748 
295.247 
294.748 
294.251 
293.756 
293 . 262 


292.7°70 
292.279 





319.623 





313.860 2 











Loga- 
rithm. 


log. R. 


2.504638 


503841 
503045 
502251 
.501459 
.500668 
.499879 
.499091 

498304 

497519 
.496736 
495953 
.495173 
.494893 
.493616 
492839 
.492064 
.491291 
.490518 
.489748 


2.488978 
-488210 
487444 
.486679 
485915 
-485152 
.484391 
.483632 
482873 
.482116 

2.481361 
480607 
479854 
479102 
478352 
477603 
476855 
.476109 
475364 
.474621 


2.473878 
473137 
472398 
471659 
470922 
470186 
.469452 
.468718 
.467986 
467256 


2.466526 


2.459300 








Tang. 
Off. 


t. 








Mid. 
Ord. 
m. 





8.935 | | 


3.942 
3.950 
3.957 
3.964 
3.972 
3.979 
3.986 
3.994 
4.001 


4.008 
4.016 
4.023 
4.030 
4.038 
4.045 
4.052 
4.060 


4.067 || 
4.074 || 


4.081 
4.089 
4.096 
4.108 
4.111 


4.118 || 


4.125 
4.133 
4.140 


3} 4.147 
4,155 || 


4.162 
4.169 


4.177 || 














: Loga- | Tang. 

Deg. Radius.) vitnm, | Off. 

D. R. / log. R. te 
20° 0’) 287.939 2.459300) 17.365 
10 | 285.583) .455733! 17.508 

20 | 283.267} .452195! 17.651 

30 | 280.988 | .448688) 17.794 

40 | 278.746 | .445209| 17.987 

50 | 276.541 | .441759 18.081 
21° 0’) 274.370 |2. 488337) 18.224 
10 | 272.234 | .434943! 18.367 

20 | 270.132] .431576; 18.509 

30 | 268.062 | .428235 18.652 

40 | 266.024 | 7424921) 18.795 

50 | 264.018 | .421633) 18.938 
22° 0’ 262.042 |2.418371) 19.081 
10 | 260.098 | .415134! 19.224 

20 | 258.180 | .411922) 19.366 

30 | 256.292 | .408734| 19.509 

40 | 254.431} 1405571) 19.652 

50 | 252.599 | .402431| 19.794 
23° 0/| 250.793 |2.399315! 19.937 
10 | 249.013 | .396222) 20.079 

20 | 247.258) .393151) 20.222 

30 | 245.529) .390103) 20.364 

40 | 243.825 | .88707'7, 20.507 

50 | 242.144] .384074/ 20.649 

| 24° 0’) 240.487 2.381091) 20.791 
10 | 238.853 | .378130 20.933 
20 | 237.241 | .875190 21.076 

30 | 235.652 | .372270) 21.218 
40 | 234.084) .369371) 21.360 

50 | 232.5371 .366492 21.502 
25° 0’| 231.011 |2.363633 21.644 
10 | 229.506 | .860794 21.786 
20 | 228.020} .357974) 21.928 | 
30 | 226.555 | .355173; 22.070. 

40 | 225.108 | .352391; 22.212 
50 223.680 | .349627 22.353 
26° 0’) 222.271 '2.846882) 22.495 
10 | 220.879| .344155 22.637 
20 | 219.506 | .£41446 22.778 | 

30 | 218.150 | .338755) 22.920 
40 | 216.811 "336081, 23.062 | 

50 | 215.489 | .333424 23.203 
27° 0/| 214.183 |2.330785) 23.345 
10 | 212.893 | .328162 23.486 

20 | 211.620 | .825556) 23.627 
30:| 210.362] .322967 23.769 

40 209.119 | .320393 28.910 

50 | 207.891 | .317836 24.051 
28° 0’) 206.678 |2.815295 24.192 
10 | 205.480 | .312769 24.333 

20 | 204.296 | .810259, 24.474 

30 | 203.125 | .307764' 24.615 
40 | 201.969 "305285 | 2 756 

50 | 200.826 | .302820 24.897 
29° 0/| 199.696 |2.300370) 25.038 
10) 198.580| .297935 25.179 

20 | 197.476 | .295515) 25.320 

30 | 196.885 | .293108) 25.460 

40 | 195.306} .290716 25.601 

50 | 194.240] 1288338, 25.741 
30° 0/| 193.185 |}2.285974! 25.882 


TABLE I.—RADII, LOGARITHMS, OFFSETS, ETC. 

















Mid. 
Ord. 
p Imm, 








TABLE I.—RADII, LOGARITHMS, OFFSETS, ETC. 9 



































| | 
| . Loga- | Tang..| Mid. : Loga- | Tang. | Mid. 
Deg. Radius.) rithm, | Off. | Ord. || Pes: Radius.) rithm, | om. | Ord. 
D. R. log. R. te me D. R, log. R. t. Me 
| | | 
830° 20’ 191.111 |2.281286 | 26.163 | 6.657 || 38°30’) 151.657/2.180863 32.969} 8.479 
40 | 189.083| .276652 | 26.443 | 6.731 || 89° 0’ 149.787] .175475 33.381 | 8.592 
31° 0 | 187.099} .272071 | 26.724 | 6.805 || 30 | 147.965) .170160 33.792] 8.704 
20 | 185.158! .267541 | 27.004 | 6.879 || 40° 0’| 146.190} .164918 34.202] 8.816 
40 | 183.258} .263062 | 27.284 | 6.958 30 | 144.460] .159747 34.612] 8.929 
32° 0’ 181.398) .258632 | 27.564 | 7.027) 41° 0’) 142.773) .154645 35.021 | 9.041 
20 | 179.577 | .254250 | 27.843 | 7.101 30 | 141.127] .149610 35.429 | 9.154 
40 | 177.794| .249916 | 28.123 | 7.175 || 42° 0’, 139.521! .144641 35.887 | 9.267 
33° 0/| 176.047 | .245628 | 28.402 | 7.250 30 | 137.955! .189736 36.244] 9.380 
20 | 174.336 | .241386 | 28.680 | 7.3824 || 48° 0’ 186.425} .184895 36.650| 9.493 
40 | 172.659 | .237188 | 28.959 | 7.398. 30 | 134.932] .130114 37.056 | 9.606 
34° 0/| 171.015 |2. 233035 | 29.287 | 7.473 || 44° 0” 133.473)2 .125895 37.461 | 9.719 
20 | 169.404 | .228924 | 29.515 | 7.547 30 | 132.049} .120734 37.865 | 9.832 
40 | 167.825 | .224855 | 29.793 | 7.621 || 45° 0’ 130.656) .116180 38.268] 9.946 
35° 0’| 166.275} .220828 | 30.071 | 7.696 30 | 129.296) .111584 38.671 |10.059 
- 20 | 164.756 | .216842 | 30.348; 7.'770|| 46° 0’, 127.965] .107092 39.073 |10.173 
40 | 163.266] .212895 | 30.625 | 7.845 30 | 126.664] .102655 39.474 |10.286 
86° 0’| 161.803} .208988 | 30.902 | 7.919 || 47° 0’, 125.392] .098270 39.875 |10.400 
20 | 160.368] .205119 | 31.178 | 7.994 30 | 124.148] .093938 40.275 |10.516 
40 | 158.960; .201288 | 31.454 | 8 068|| 48° 0’ 122.930) .089657 40.674 |10.628 
87° 0’| 157.577| .197494 | 31.730] 8.143 30 | 121.738] .085425 41.072 |10.742 
20 | 156.220] .193736 | 82 006 | 8.218 || 49° 0’, 120.571| .081243 41.469 |10.856 
40 | 154.887] .190014 | 32.282 | 8.292 30 119.429) .077109 41.866 |10.970 
88° 0’| 153.578 |2.186328 | 32.557 | 8.367 || 50° 0’) 118.310)2.073022) 42.262 |11.085 








' TABLE II.— CORRECTIONS FOR TANGENTS AND EXTERNALS 





For TANGENTS, ADD For EXTERNALS, ADD 








Ang} 5° | 10° | 15° | 20° | 25° | 30° |/Amg| 5° | 10° | 15° | 20° | 25° | 30° 
4 |Cur.| Cur.) Cur.| Cur.| Cur.) Cur.|| © | Cur.| Cur.| Cur. | Cur. | Cur. | Cur. 

































































. . . . ie 1 
60 | .21 | .42 | .638) .84/1.05)1.27)) 60 | .056| .112| .168| .225 | .283 | .340 
70 | .25 | .51 | .76) 1.02 | 1.28)1.54}; 70 | .080| .159| .240). .821| .403| .485 
80 | .80 | .61 | .91 | 1.22)1.53)1.84]| 80 | .110| .220| .832) .445} .558) .671 
90 | .86 | .72 | 1.09 | 1.45 | 1.83 | 2.20|| 90 | 149] .299| .450| .608| .756 | .910 
100 | .43 | .86 | 1.30) 1.74| 2.18} 2.62||100 | .200| .401| .604| .809 |1.015 )1.221 
110 | .51 |1.08 | 1.56 | 2 08 | 2.61 | 3.14}/110 | .268| .536| .806 /1.082 |1.355 |1.683 
120 | .62 [1.25 | 1.93 | 2.52 | 3.16 | 3.81 ||120 | .360]-.721 |1 .086 1.456 |1.825 |2.197 





10 











TABLE III.—TANGENTS AND EXTERNALS 

















12 


13 


14 


15 


16 


17 


18 


19 


20 




















22 


23 


24 


25 


26 


27 


28 


29 





30 




















TO A 1° CURVE 


11 








32 


33 


34 


35 


36 


37 


38 


39 


40 


























216.25 
218.66 
221.08 
223.51 
225.96 
228.42 
230.90 
233.39 
235.90 
238.43 
240.96 
243.52 


246.08 
248.66 
251.26 
253.87 
256.50 
259.14 
261.80 
264.47 
267.16 


269.86 


272.58 
275.31 
278.05 
280.82 
283.60 
286.39 
289.20 
292.02 


294.86 


297.72 
300.59 
303.47 
306.37 
809.29 


312.22 


815.17 | 








318.13 


321.11 
324.11 
827.12 
330.15 
333.19 
336.25 
339.32 


342.41 | 


345.52 


348.64 
351.78 
804. 94 
358.11 
861.29 
864.50 


367 #22 | 
370.95 | 


374.20 


877.47 
. 880.76 


384.06 








42 


43 


44 


45 


46 


47 


48 


49 


50 











Tan- | Exter- 
gent. nal. 

yi EK. 
2142.2 | 387.38 
2151.7 | 390.71 
2161.2 | 394.06 
2170.8 | 397.48 
2180.3 | 400.82 
2189.9 | 404.22 
2199.4 | 407.64 
2209.0 | 411.07 
2218.6 | 414.52 
2228:1 | 417.99 
2237.7 | 421.48 
2247.3 | 424.98 
2257.0 | 428.50 
2266.6 | 482.04 
2276.2 | 435.59 
2285.9 | 439.16 
2295.6 | 442.75 
2305.2 | 446.35 
2314.9 | 449.98 
2324.6 | 453.62 
2384.3 | 457.27 
2344.1 | 460.95 
2353.8 | 464.64 
2363.5 | 468.35 
2373.3 | 472.08 
2383.1 | 475.82 
2392.8 | 479.59 
2402.6 | 483.37 
2412,4 | 487.17 
2422.3 | 490.98 
2432.1 | 494.82 
2441.9 | 498.67 
2451.8 | 502.54 
2461.7 | 506.42 
2471.5 | 510.33 
2481.4 | 514.25 
2491.3 | 518.20 
2501.2 | 522.16 
2511.2 | 526.13 
2521.1 | 530.13 
2581.1 | 584.15 
2541.0 | 588.18 
2051.0 | 542.23 
2561.0 | 546.30 
2571.0 | 550.39 
2581.0 | 554.50 
2591.1 | 558.63 
2601.1 | 562.7 
2611.2 | 566.94 
2621.2 | 571.12 
2631.3 | 575.382 
2641.4 | 579.54 
2651.5 | 583.78 
2661.6 | 588.04 
2671.8 | 592.32 
2681.9 | 596.62 
2692.1 | 600.93 
2702.3 | 605.27 
2712.5 | 609.62 
2722.7 | 614.00 











51° 


53 


54 


55 


56 


57 


58 


59 


60 





Tan- 
gent, 


T. 





2732.9 
2743.1 
2753.4 
2763.7 
2773.9 
2784.2 

794.5 
2804.9 
2815.2 
2825.6 
2835.9 
2846.3 


2856.7 
2867.1 
2877.5 
2888.0 
2898.4 
2908.9 
2919.4 
2929.9 
2940.4 
2951.0 
2961.5 


+ 2072.1 


2982.7 
2993.3 
8003.9 
8014.5 
8025 .2 
3035.8 
8046.5 
3057.2 
3067.9 
8078.7 
8089 4 
8100.2 


3110.9 
8121.7 
8132.6 
8143.4 
3154.2 
3165.1 
3176.0 
3186.9 
3197.8 
8208.8 
3219.7 
8230.7 


8241.7 


eee 


Exter- 
nal, 


E. 


ae .66 

7.32 
681.09 
686.68 
691.40 
696.18 
700.89 
705.66 
710.46 
715.28 
720.11 
724.97 


729.85 
734.76 
739.68 
744.62 
749.59 
754.57 
759.58 





12 


TABLE III.—TANGENTS AND EXTERNALS 








62 


63 


64 


65 


66 


67 


68 


69 


70 


























1134.8 


1141.4 
1148.0 
1154.7 
1161.3 
1168.1 
1174.8 
1181.6 
1188.4 
1195.2 
1202.0 
1208.9 
1215.8 


1222.7 
1229.7 
1236.7 
1243.7 
1250.8 
1257.9 
1265.0 
1272.1 
1279.8 
1286.5 
1298.6 
1300.9 








73 








72 


75 


76 


17 


78 





79 


80 


























82 


83 


85 


_ 86 


87 


88 


89 


Tan- 
gent. 


T. 


4893 .6 
4908.0 
4922.5 
4987.0 
4951.5 
4966.1 
4980.7 
4995 .4 
5010.0 
5024.8 
5089.5 
5054.3 


5069.2 
5084.0 
5099.6 
5113.9 
5128.9 
| 5143.9 

5159.0 


5204.4 
| 5219.7 
| 5284.9 
5250.3 
5265.6 
5281.0 
5296.4 
5311.9 
5827.4 

5343.0 
| 5358.6 
5374.2 
5389.9 
5405.6 
5421.4 


5437.2 
5453.1 











; 5174.1 
| 9189.3 





Exter- 
nal. 


EK. 


1805.3 
1814.7 
1824.1 
1833.6 - 
1843.1 
1852.6: 
1862.2 
1871.8 
1881.5 
1891.2 
1900.9 
1910.7 


1920.5 
1930.4 
1940.3 
1950.3 
1960.2 
1970.3 
1980.4 
1990.5 
2000.6 
2010.8 
2021.1 
2081.4 


2041.7 
2052.1 
2062.5 
2073.0 
2088.5 
2094.1 
2104.7 
2115.3 
2126.0 
2136.7 
2147.5 
2158.4 


2169.2 


| 2180.2 
| 2191.1 


2202.2 
2218.2 


| 2224.3 
| 2235.5 


2246.7 
2258.0 


| 2269.3 


2280.6 


\ 2292.0 


2803.5 
2315.0 
2326.6 
2388 .2 
2349.8 





TO A 1° CURVE 13 

































































Tan- Ex- Tan- Ex- -| Tan- Ex- 
Angle. gent. -| ternal. |; Angle gent. | ternal. Angle. gent. | ternal 
A Ts E. A a E. A ARS EK. 

O15 5830.5 | 2444.9 |} 101° 6950.6 | 3278.1 |; 111° 8336.7 | 4386.1 
10’| 5847.5 | 2457.1 10’| 6971.3 | 3294.1 10’| 8362.7 | 4407.6 

20 | 5864.6 | 2469.3 20 | 6992.0 | 3310.1 20 | 8388.9 | 4429.2 

380 |°5881.7 | 2481.5 30 | 7012.7 | 3326.1 30 | 8415.1 | 4450.9 

40 | 5898.8+| 2493.8 40 | 7033.6 | 3342.3 40 | 8441.5 | 4472.7 

50 | 5916.0 | 2506.1 50 | 7054.5 | 3358.5 50 | 8468.0 | 4494.6 

92 5933.2 | 2518.5 || 102 7075.5 | 3374.9 |} 112 8494.6 | 4516.6 
10 | 5950.5 | 2531.0 10 | 7096.6 | 3391.2 10 | 8521.3 | 4538.8 

20 | 5967.9 | 2543.5 20 | 7117.8 | 3407.7 >) 20°] 8548-1 | 4561-1 

30 | 5985.3 | 2556.0 380 | 7189.0 | 3424.3 30 | 8575.0 | 4583.4 

40 | 6002.7 | 2568.6 40 | 7160.8 | 3440.9 40 | 8602.1 | 4606.0 

50.| 6020.2 | 2581.3 50 | 7181.7 | 3457.6 50 | 8629.3 | 4628.6 

93 6037.8 | 2594.0 || 103 7203.2 | 3474.4 || 118 8656.6 | 4651.3 
. 10} 6055.4 | 2606.8 10 | 7224.7 | 3491.3 10 | 8684.0 | 4674.2 
20 | 6073.1 | 2619.7 20 | 7246.3 | 3508.2 20 | 8711.5 | 4697.2 

30 | 6090.8 | 2632.6 30 | 7268.0 | 3525.2 30 | 8739. 2 4720.3 

40 | 6108.6 | 2645.5 40 | 7289.8 | 3542.4 40 | 8767.0 | 4743.6 

50 | 6126.4 | 2658.5 50 | “7311-7 |: 8559.6 50 | 8794.9 | 4766.9 
94 6144.3 | 2671.6 || 104 7333.6 | 3576.8 || 114 8822.9 | 4790.4 
10 | 6162.2 | 2684.7 10 | 7355.6 | 3594.2 10 | 8851.0 | 4814.1 

20 | 6180.2 | 2697.9 20 (30028 | 861127 20 | 8879 4837.8 

30 |. 6198.3 | 2711.2 30 | 7399.9 | 3629.2 30 | 8907. 4861.7 

40 | 6216.4 | 2724.5 40 | 7422.2 | 3646.8 40 | 8936. 3 4885.7 

50 | 6234.6 | 2737.9 50 | 7444.6 | 3664.5 _ 50} 8965.0 | 4909.9 

95 6252.8 | 2751.3 || 105 7467.0 | 3682.3 || 115 8993.8 | 4934.1 
10 | 6271.1 | 2764.8 10 | 7489.6 | 3700.2 10 | 9022.7 | 4958.6 

20 | 6289.4 | 2778.3 20 | 7512.2 | 3718.2 20 | 9051.7 | 4983.1 

30 | 6307.9 | 2792.0 30 | 7534.9 | 3736.2 30 | 9080.9 | 5007.8 

40 | 6326.3 | 2805.6 40 | 7557.7 | 3754.4 40 | 9110.3 | 5082.6 

50 | 6344.8 | 2819.4 50 | 7580.5 | 3772.6 50 | 9139.8 | 5057.6 
96 6363.4 | 2833.2 || 106 7603.5 | 8791.0 || 116 9169.4 | 5082.7 
10 | 6382.1 | 2847.0 10-) 7626.6 | 8809.4 10 | 9199.1 | 5107.9 

20 | 6400.8 | 2861.0 20 | 7649.7 | 8827.9 20 | 9229.0 | 5133.3 

30 | 6419.5 | 2875.0 30 | 7672.9 | 3846.5 30 | 9259.0 | 5158.8 

40 | 6438.4 | 2889.0 40 | 7696.3 | 3865.2 40 | 9289.2 | 5184.5 

50 | 6457.3 | 2908.1 50 | 7719.7 | 8884.0 50 | 9819.5 | 5210.3 

97 6476.2 | 2917.3 || 107 "743.2 | 3902.9 || 117 9349.9 | 5236.2 
10 | 6495.2 | 2931.6 10 | 7766.8 | 3921.9 10 | 9380.5 | 5262.3 

20 | 6514.3-| 2945.9 20 | 7790.5 | 3940.9 20 | 9411.3 | 5288.6 

30 | 6533.4 | 2960.3 30 | 7814.3 | 3960.1 80 | 9442.2 | 5315.0 

40 | 6552.6 | 2974.7 40 | 7838.1 | 3979.4 40 | 9478.2 | 5841.5 

50 | 6571.9 | 2989.2 50 | 7862.1 | 3998.7 50 | 9504.4 | 5868.2 

98 6591.2 3003.8 || 108 "7886.2 | 4018.2 |} 118 9535.7 | 5395.1 
10 | 6610.6 | 3018.4 10 | 7910.4 | 4037.8 10 | 9567.2 | 5422.1 

20 | 6630.1 | 3033.1 20 | 7934.6 | 4057.4 20 | 9598.9 | 5449.2 

30 | 6649.6 | 3047.9 80 | 7959.0 | 4077.2 380 | 9680.7 | 5476.5 
40°) 6669.2 | 3062.8 40 | 7983.5 | 4097.1 40 | 9662.6 | 5504.0 

50 | 6688.8 | 3077.7 50 | 8008.0 | 4117.0 50 | 9694.7 | 5531.7 

99 708.6 | 3092.7 || 109 8032.7 | 4187.1 || 119 9727.0 | 5559.4 
10 | 6728.4 | 3107.7 10 | 8057.4 | 4157.3 10 | 9759.4 | 5587.4 

20 | 6748.2 | 3122.9 20 | 8082.3 | 4177.5 20 | 9792.0 | 5615.5 

380 | 6768.1 | 3138.1 30 | 8107.3 | 4197.9 80 | $824.8 | 5643.8 

40 | 6788.1 | 3153.3 40 | 8182.3 | 4218.4 40 | 9857.7 | 5672.3 

50 | 6808.2 | 3168.7 50 | 8157.5 | 4239.0 50 | 9890.8 | 57C0.9 
100 6828.3 | 3184.1 || 110 8182.8 | 4259.7 || 120 9924.0 | 5729.7 
10 | 6848.5 | 3199.6 10 | 8208.2 | 4280.5 10 | 9957.5 | 5758.6 

20 | 6868.8 | 3215.1 20 | 8288-7 | 4801/4 | 20 | 9991.0 | 5787.7 

30 | 6889.2 | 3230.8 Va 8259.3 | 4822.4 830 |10025.0 | 5817.0 

40 | 6909.6 | 8246.5 8285.0 | 4343.6 40 |10059.0 | 5846.5 

50 | 6930.1 | 8262.3 8310.8 | 4364.8 50 |10093.0 | 5876.1 


EES 


r 





100.020 
022 











TABLE IV.-—LONG CHORDS 


2 
Stations. 


200 000 
199.999 
199.998 
199.997 
199.995 
199.992 
199.990 
199.986 
199.983 
199.979 
199.974 


199.970 
199.964 
199.959 
199.952 
199.946 
199.939 
199.931 
199.924 
199.915 
199.907 
199.898 
199.888 
199.878 
199.868 
199.857 


199.756 
199.741 
199.726 
199.710 
199.695 
199.678 
199.662 
199.644 
199.627 
199.609 
199.591 
199.572 
199.553 
199.533 


199.518 
199.492 
199.471 
199.450 
199.428 
199.406 
199.383 
199.360 
199.337 
199.313 
199.289 
199.264 
199.239 














LonG CHORDS. 





3 
Stations. 


299.999 
299.997 
299.992 
299.986 
299.979 
299,970 
299 . 959 
299 . 946 
299 . 932 
299.915 
299.898 


299.878 
299.857 
299 .834 
299.810 
299.783 
299.756 
299 726 
299.695 
299 . 662 
299 .627 
299.591 
299.553 
299.513 
299.471 
299 .428 
299.383 
299.337 
299 289 
299 239 
299.187 
299 .184- 
299.079 
299.023 
298. 964 


298 . 904 
298.843 
298.779 
298.714 
298 , 648 
298.579 
298.509 
298 .438 
298 . 364 
298.289 
298.212 
298.134 


298 .054 
297.972 
297 .888 


297.062 - 


296 962 





4 
Stations. 


899 998 
399.992 
399.981 
899.966 
399.947 
399 . 924 
399 .896 
399.865 
899.829 
899 . 789 
899. 744 
399.695 
899.643 
899.586 
899 . 524 
399 .459 
399 7389 
899.315 
399.237 
399.154. 
399.068 
398.977 
898 . 882 
398.782 
398.679 
398 571 
398.459 


895.142 
894.938 
394.731 
894.518 
394.802 
894.082 
393.857 
393.629 
393.3896 
393.159 
392.918 
392.673 
392.424 

















5 


Stations. 


499 .996 
499 .983 
499 . 962 
499 932 
499.894 
499 848 
499.793 
499.729 
499.657 
499 577 
499.488 


499 .391 
499 285 
499.171 
499 049 
498.918 
498.778 
498 . 630 
498 .474 
498 .309 
498 . 136 
497.955 
497.765 
497 .566 
497 .360 
A497 .145 
496 .921 
496 .689 
496.449 
496 .201 
495 .944 
495.678 
495 .405 
495.123 
494.832 
494.534 
494 227 
493 .912 
493 .588 
493.257 
492.917 
492.568 
492.212 
491.847 


491.474 . 


491.093 
490.704 


490.306 
489 .900 
489 .486 
489 064 
488 . 634 
488.196 
487.749 
487 .294 
486.832 
486 . 361 
485 . 882 
485.3895 
484.900 


6 
Stations. 


599 .993 
599.970 
599.933 
599.882 
599.815 
599.733 
599. 637 
599.526 
599.401 
599.260 
599.105 


598.934 
598. 750 
598.550 
598 .336 
598.106 
597.862 
597.604 
597.331 
597.043 
596.740 
596 .423 
596.091 


AD sigan geek 5 TS ee |: Ret ASE 








7 
Stations. 





699.988 
699.953 
699.893 
699.810 
699.704 
699 574 
699 .420 
699 .242 
699.041 
698.816 
698 .567 


698.295 
698 . 000 
697.680 
697 .338 
696.971 
696.581 
696.168 
695.731 
695.271 
694.787 
694.280 
693.750 | 


693.196 
692.619 
692.018 
691.395 
690.748 
690.079 
689.386 
688 . 670 
687 .930 
687.169 
686 .384 
685 .576~ 


684.745 
683.892 
683.016 
682.117 
681.195 
680,251 
679.285 
oes 296 

7.284 
or6. 250 
675.194 
674.116 


673.015 
671.892 
670.748 
669.581 


TABLE IV.—LONG CHORDS 


Lona CHORDS. 





$8 
Stations. 





799.982 
799.929 
799.840 
799.716 
799.556 
799 . 360 
799.130 
798.863 
798.562 
798 . 224 
(97.852 


797.444 
797 000 
796 .522 
796.008 
795.459 
794.874 
994.255 
793.600 
792.911 
792.186 
791.427 
790.6382 
789 .803 


788 . 939 
788 .040 





761.309 


759.670 
"57.999 
"56. 295 
754.560 
752.792 
950.998 
749.161 
_ 47.209 
45.404 
743.479 
741.522 
(39.535 
737.516 





9 
Stations, 


899.974 

899.899 
899.772 
899.594 
899.365 
899.086 
898 . 757 
898 .376 


869.219 


867.454 
865.642 
863.782 
861.875 

859.922 
857.921 


811.314 








10 
Stations. 


999 . 965 
999. 860 
999. 686 
999.442 
999. a 
998 .7 

998 . 590 
997. bi 
997.175 
996.518 
995. 782 


994.981 
994.112 
993.173 








11 
Stations. 


1099.95 
1099.81 
1099.58 
1099.25 
1098 . 84 
1098.33 
1097.72 
1097.02 
1096.23 
1095.35 
1094.38 


1093.31 
1092.15 
1090.90 
1089.56 
1088 . 12 
1086.60 
1084.98 
1083.28 
1081.48 
1079.59 
1077.61 
in 5.54 


073.38 
s0eL 14 
1068.81 
1066.38 
1068 .87 

1061.27 
1058 .59 
1055.81 
1052.95 
1050.01 
1046.97 
1043.86 


1040.66 
1087.37 
1034.01 
1080.55 
1027 .02 
1023.40 
1019.70 
1015.93 
1012.07 
1008.18 
1004.11 
1000.01 


995 .834 
991.580 
987. 250 


944,933 
939.871 














15 


12 
Stations. 


1199.94 
1199.76 
1199.46 
1199.08 
1198.49 
1197.82 
1197.04 
1196.18 
1195.11 
1193.96 
1192.69 


1191.31 
1189.80 
1188.18 
1186.48 
1184.57 
1182.59 
1180.49 
1178.28 
1175.94 
1173.49 
1170.93 
1168.25 


1165.45 
1162.54 
1159.51 
1156.37 
1153.12 
1149.76 
1146.28 
1142.69 
1188.99 
1185.18 
1181.26 
1127.24 


1123.10 
1118.86 
1114.51 
1110.05 
1105.49 
1100.83 
1096.06 
1091.19 
1086 .22 
1081.15 
1075.98 
1070.71 


1065.34 
1059.88 
1054.32 
1048.66 
1042.91 
1037.06 
1031.13 
1025.11 
1018.99 
1012.79 
1006.49 
1000.12 
993.658 


11 


12 


13 


14 


15 


16; 


17 


18 


19 


20 














Stations. 


199.213 
199.187 
199.161 
199.1384 
199.107 
199.079 
199.051 
199.023 
198 . 994 
198. 964 
198.935 


198 . 904 
198.874 
198.843 
198.811 
198.779 
198.747 
198.714 
198.681 
198.648 
198.614 
198.579 
198.544 


198.509 
198.474 
198.437 
198.401 
198 .364 
198.827 
198 . 289 
198.251 
198.212 
198.173 
198.134 
198 .094 


198.054 
198.013 
197.972 
197.930 
197.888 
197.846 
197.803 
197.760 


197.716 © 


197.672 
197.628 
197.583 


197.538 
197.492 
197.446 
197.399 
197.352 
197,305 
197.256 
197.209 
197.160 
197.111 
197.062 
197.012 
196.962 














Lona CHORDS. 


3 
Stations. 


296 .860 
296.756 
296.651 
296.544 
296 .436 
296.325 
296.214 
296.100 
295 . 985 
295.868 
295.750 


295 . 629 
295.508 
295 . 384 
295 .259 
295.182 
295 . 004 
294.874 
294.742 
294.609 
294.474 
294.337 
294.199 


294.059 
293.918 
293.774 
293.629 
293.483 
293.335 
293.185 
293 .034 
292.881 
292.726 
292.570 
292.412 


292. 252 
292.091 
291 .928 
291.764 
291 .598 
291.480 
291.261 
291.090 
290.918 
290.743 
290.568 
290.390 





4 


Stations. 


392.171 
391.914 
391.652 


389 .'708 
389.414 


389.116 
388 .814 
388 . 508 
388.197 
387.883 
387.565 
887.248 
386.916 
386 .586 
386 . 252 
385.914 
385.572 


385.225 


376.615 
376.179 


375.739 
375.295 
874.848 
374.397 
873.942 
873.483 
373.021 
372.554 
372.084 
371.610 
871.183 
370.652 
370.167 





TABLE IV.—-LONG CHORDS 


5 


Stations. 


484.397 
483 .886 
483 .367 
482.840 
482.305 
481.762 
481.211 
480.653 
480.086 
479 511 
478 . 929 


478 .338 
477.740 
477.1385 
476.521 
475 .899 
475.270 
474.633 
473 .988 
473 2336 
472.675 
472.007 
471 .332 


470.649 
469.958 
469 .260 
468 .554 
467 .840 
467.119 
466.390 
465.654 
464.911 
464.160 
463.401 
462.635 


461.862 
461.081 
460.293 
459 498 
458 .695 
457 .886 
457.069 
456.244 
455.413 
454.574 
453.728 
452.875 


452.015 
451.147 








6 
Stations, 





572.813 
571.926 
571.027 
570.113 
569.186 
568.245 
567.292 
566.824 
565.343 
564.349 
563.341 


562.321 
» 561 287 
560.240 
559.180 
558.107 
557.026 
555.921 
554.809 
553.684 
5d2.546 
551.395 
550.232 
549 .056 
547.867 
546 .666 
545 .452 
544.226 
542.987 
541.736 
540.472 
539.196 
537.908 
536 .608 
535.296 


517.160 
515.685 
514.198 
512.699 
511.190 





TABLE IV.—LONG CHORDS We 





Lone CHORDS. 























Degree 
of 
Curve. 7 & 9 10 11 12 
Stations.| Stations. Stations. | Stations. | Stations. | Stations. 
10° 10 | 656.723 735.467 808 . 426 875.025 934.741 987.105 
20 | 655.320 733.387 805.495 871.058 929 .542 980.473 
380 | 653.895 731.277 802.524 867.038 924.276 973.760 
40 | 652.450 729 137 799 512 862.963 918.943 966.967 
50 | 650:983 726 . 967 796.458 858.836 913.544 960.093 
ab ‘| 649.496 724 767 793.364 854.656 908 . 080 953.141 
10 | 647.989 722.537 790 .230 850.425 902.550 946.112 
20 | 646.460 720.278 787 .056 846.140 896.957 939 .007 
80 | 644.911 717.990 783.843 841.808 891.303 931.828 
40 | 648.342 715.672 780.590 837 .424 885.586 924.575 
50 | 641.752 713.325 777 298 832.990 879.807 917.250 
12 640.142 710.950 773.968 828 .507 873.968 909. 854. 
10 | 688.512 708.546 770.600 823.974. 868.070 902.389 
20 | 636.862 706.113 767.193 819.394 862.113 894.855 
80 | , 635.191 703.653 763.749 814.766 856.099 887 .254 
40 | 633.501 701.164 760.268 810.092 850.028 79.588 
50 | 631.792 698.647 756.749 805.370 843.900 871.857 
13 630.062 696.103 753.194 800.602 837.718 864.063 
10 | 628.313 693.531 749 603 795.790 831.482 856.208 
20 | 626.544 690. 932 45.976 790.932 825.192 848.293 
80 | 624.756 688 . 306 "42 313 786 .030 818.850 840.318 
40 | 622.949 685 .653 738.616 781.085 812.457 832.286 
50 | 621.128 682.974 734.883 776 .096 806.013 824.198 
14 619.278 | 680.268 731.116 771.066 799 .520 816.056 
10 | 617.418 677 535 "27 315 765.993 792.979 807.860 
20 | 615.530 674.777 723 .480 760.87 786.389 799.612 
380 | 613.628 671.993 719.612 WB5 .725 779.753 791.313 
40 | 611.708 669.1838 715.711 750.531 773.072 782.966 
50 | 609.769 666.348 ALS CC7 945.297 766.345 V74..571 
15 607.812 663.488 707.811 740.024 959.575 766.180 
10 | 605.836 660.603 703.814 734.714 752.763 
20 | 603.842 657 .693 699.785 729 .366 745.908 
380 | 601.831 654.758 695.725 723 .982 739.014 
40 | 599.801 651.799 691 . 634 718.561 732.078 
50.| 597.753 648.817 687.518 7137105 725 .104. 
16 595.688 645.810 683.362 07.614 718.092 
10 | 593.605 642.780 679.182 702.088 711.043 
20 | 591.505 639.727 674.973 696 .529 708.959 
30 | 589.388 636 .650 670.735 690.938 
40 | 587.253 633.550 666.469 685.314. 
50 585.101 630.428 662.175 679.659 
17 582.933 627.283 657 . 854. 673.972 
10 |. 580.747 624.117 653.506 668.256 
20 | 578.545 620.928 649.131 662.510 
80 | 576.326 617.717 644.730 656.735 
40 | 574.091 614.485 640.304 650.933 
50 | 571.839 611.232 635.852 645.103 
18 569.571 607.958 631.375 689.245 
10 | 567.287 604.664 626.874 
20 | 564.988 601.349 622.3849 
80 | 562.673 598.013 617.801 
40 | 560.342 594.658 613.229 
50 | 557.996 591 . 283 608 .635 
19 555 . 634. 587.888 604.018 
10 | 558 257s. 584.475 599.379 
20 | 550.864 581.042 594.720 
80 | 548.457 577.591 590.039 
40 | 546.0385 574.121 585 .339 
50 | 543.599 570. 634 580.618 





20 541.147 567.128 575.877 








18 TABLE IV.—LONG CHORDS 


Lone CHORDS. 

















Actual 
Degree ARG 
Corve.| «On? 2 3 4 5 6 
Station.) sidtions. | Stations: | Stations’ | Stations’ Stations! 

Zle 100.562 196.651 286.716 367.179 435.345 488. 931 
22 100.617 196.325 285. 437 364. 060 429 . 305 478.775 
23 100.675 195.985 284.101 360. 810 423. 033 468.270 
24 100.735 195.630 282.709 857. 433 416.535 457. 433 
25 100.798 195.259 281.262 | 353.930 409. 819 446.280 
26 100. 863 194. 874 279.759 390. 803 402. 891 434.827 
27 100.931 194.474 278.201 846.555 395.758 423.092 
28 101.002 194.059 276.589 342.688 “388. 428 411.092 
29 101.075 193. 630 274.924 338. 704 380.908 398.846 
30 101.152 193.185 273.205 334. 607 373.205 386.370 
31 101.230 192.726 271. 433 330. 397 365. 328 373.685 
32 101.312 192.252 269.610 326.078 857 . 284 360.808 
33 101.396 191.764 267.734 321.654 349.081 347.759 
34 101.482 191.261 . 265.808 SiTil25 340.729 334.556 
35 101.572 190.743 203.830 312.496 Soe Boe 321.220 
36 101.664 190.211 261.803 307.768 323. 607 307.768 
37 101.759 189.665 209.727: 302.946 314. 855 294.222 
38 101.857 189.104 257.602 © 298. 032 805. 987 280.600 
39 101.957 188.528 255.429 293.028 297.012 266.923 
40 102.060 187.939 253.209 287.938 287.939 253.209 
4] 102.166 187.334 250.942 282.766 278.777 239.478 
42 102.275 186.716 248.629 277.514 269. 535 225.750 
43 102.386 186.084 246.271 272.186 260. 222 212.045 
44 102.500 185.4387 243. 868 266. 784 250. 848 198.380 
45 102.617 184.776 241.421 261.313 241.421 184.776 
46 102.737 184.101 238.932 255. 775 231.952 171.251 
47 102.860 183.412 236. 400 250.173 222.448 157.824 

__ 48 102.985 182.709 233.826 244.512 212.920 144.512 
49 103.114 181.992 231.212 238.795 203.377 131.335 
50 103.245 181.262 - + 228.558 233.025 193.828 118.310 





nn eee 


TABLE V.—LINEAR DEFLECTION TABLE 


19 








30 


30 
30 
30 
30 


30 
10 
30 


eSoont oar hw w 








MIA AIR MMS 
oe 
~a 
© 


00 
loro DO mt OO OY > 
33% KSERSSSRA 


— 
S 
—~ 
3 








50.08 /100.15 
50.92/101 .84 
51.76|103.53 








124.75 
127.31 


129.86 
132.42 
134.97 
137.52 
140.07 
142.61 


145.15! 1! 


147.69 
150.23 
152.76 
155.29 











254. 
258. 


(3815 








60 | 305.52 
82,310.59 





362.35 





414.11 





889.59 
897.26 
404.91 
412.56 
420.20 
427.83 
435.46 
443 .08 
450.68 
458 . 28 
465.87 











10 














TABLE VI.—MIDDLE ORDINATES 


1 


Station. 


pe fh preheat fk feed ped bed 
ean (Sie PL ols ol, (80 bak ee Sale eee blak (Od Some) uncut eine aks) me, enone ae: 
€ co 
oS 
Ne) 


2 


Stations. 








8 
Stations. 


4. 
Stations. 


6 


5 : 
Stations. | Stations. 





a ey ee ny 





TABLE VI.—MIDDLE ORDINATES 


21 





10 





7 


Stations. 


72.037 
73.744 
75.446 
77.145 
78.840 
80.531 
82.218 


83.901 
85.580 
87.254 
88. 924 
90.590 
92.252 
93.909 
95.561 
97.208 
98.851 
100.489 
102.122 
103.750 


Stations. 


108.916 
111.067 
113.210 
115 .346 
117.475 
119.596 
121.709 
423.814 
125.911 
128.000 
130.081 
132.153 
134.217 





9 


Stations. 


104.129 
106.912 
109. 686 
112.452 
115.208 
117.954 
120.691 
123.417 
126.134 
128.840 
131.535 
134.219 


136.893 
139.555 
142.205 
144.844 
147.470 
150.085 
152.687 
155.277 
157.854 
160.417 
162.968 
165.505 
168.029 





10 


Stations. 


127.995 
131.384 
134.759 
138.120 
141.468 
144.800 
148.118 
151.421 
154.708 
157.979 
161.234 
164.473 


167.695 
170.899 
174.086 
177.255 
180.407 
183.589 
186.653 
189.748 
192.824 
195.880 
198.916 
201 . 932 
204.928 


11 


Stations. 


104.323 
108 .558 
112.779 
116.986 
121.178 
125 . 356 
129.517 
133.663 


1s¢i79l 


141.903 
145.997 
150.072 


154.129 
158.166 
162.184 
166.182 
170.159 
174.114 
178 .048 
181.960 
185.850 
189.716 
198.559 
197.377 


201.171 
204.941 
208 . 685 
212.403 
216.095 
219.760 
223.898 
227.008 
230.591 
234.145 
237.670 
241.167 
244.633 


12 
Stations. 


123.862 
128.864 
133.847 
138.810 
143.753 
148.674 
153.572 
158.448 
163.300 
168.128 
172.981 
177.708 


182.459 


282.779 


287.160 
241.507 
245.818 


~ 250.093 


254.331 
258.531 
262.694 
266.818 
270.904 
274.949 
278.955 
282.919 
286.843 





11 


12 


13 


14 


15 


16 


17 


18 


19 


20 














1 


Station. 








2 
Stations. 


3 
Stations. 


38.266 
38.576 





4 
Stations. 


| 











TABLE VI.—MIDDLE ORDINATES 


5 Ee 5: 
Stations. | Stations. 





103.079 
104. 282 


105.381 
106.527 
107.669 
108 . 807 
109.941 
11107 
112.197 
113.319 
114.488 
115.552 
116.662 
117.768 


118.870 
119.967 
121.061 
122.150 
128.235 
124.315 
125.391 
126 .463 
127.530 
128.593 
129.651 
180.704 
131.753 
182.797 
133.837 
184.872 
135.902 
136 .928 
137.948 
188 .964 
189.975 
140.981 
141.982 
142.978 
148.969 








TABLE VI.—MIDDLE ORDINATES 23 











eh a | 2 3 4. 5 6 
of Station. | Stations. | Stations. | Stations. | Stations. | Stations. 
Curve 
21° 4.594 18.224 40.431 70.4738 107.344 149.809 
22 4.814 19.081 42.275 48.545 111.741 155.460 
23 5.0385 19.937 44,108 [6.577 116.042 160.917 
24 5.255 20.791 45 .929 79.570 120.243 166.172 
25 5.476 21.644 47 738 82.520 124,342 171.221 
26 5.697 22 495 49 5384 85.427 128.335 176.058 
QF 5.918 23.845 51.317 88.289 182.219 180.677 
28 6.139 24.192 53.086 91.105 135.990 185.075 
29 6.360 25 .088 54,842 93.873 139.647 189.245 
30 6.583 25 .882 56.583 96.593 143.185 193.185 
ole 6.805 26.724 58.809 99 .261 146.608 
32 ? 027 27 564 60.019- 101.878 149.898 
33 7.250 28.402 61.714 104.442 153 .068 
34 7A 29 237 63.392 106.952 156.110 
35 7.696 30.071 65.053 109.406 159 .023 
_ 36 7.919 30.902 66.698 111.8038 161.803 
37 8.143 31.730 68.825 114.1438 
38 8.367 32.557 69.933 116.424 
39 8.592 33.381 71.524 118.645 ; 
40 8.816 34,202 73.095 120.805 
41° 9.041 85,021 74.647 122.902 
42 9 267 85.837 76.180 124,937 
43 9.493 36.650 77.693 126.909 
44 9.719 37.461 79.185 128.815 
45 9.946 38.268 80.656 130.656 
46 10.173 39.073 82.107 
47 10.400 89.875 83.535 
48 10.628 40.674 84.942 
49 10.856 41.469 86 .327 


50° 11.085 42.262 87.689 





CORRECTIONS FOR SUBCHORD LENGTHS 


ee | | [| | —— | | | 





24 


TABLE VII.—COEFFICIENTS FOR VALVOID ARCS 





t 
I—RATIO OF U = 3 





IiL—RatTio oF 1, = 























80° 











, 


40° 





.3510 
3430 
3393 
0373 
3361 
33053 
3348 


)} 3344 


3399 
3385 
3331 


50° 


3506) . 
3426). 
3389) . 
3369) . 
3357). 
3349}. 
3344) . 
3340). 
3336) . 
.3331}. 
3328} . 





II.—RATIO oF 











100° 














1 








60° 





7432} . 
. 7341 | . 
7300} . 
7278 | . 
. 7265 | . 
257 |. 
7251}. 
(247). 
7241). 
(287 |. 





(234 | . 





80° | 90° | 100° 





7218 | .7090} . 














To A CHANGE OF ONE DEGREE IN THE ANGLE A. 








| 8 
S 
° 


© OF dS OV C9 0D 
AIovroorwho 








OIODore Ww or ° 
SBSSasEs | 
BH FS OT OTB 09 0D 





or 
So 
° 





O12 CO SF 0 














110° | 


110° 





6795 
6714 
.6678 
.6659 
.6648 
.6640 
6635 
6632 
6627 
6624 
6621 





120° 








120° 





6630 


Fai aa = LENGTH OF VALVOID ARC CORRESPONDING 








CB =F Od OT CODD 
SCwWWwUTtOoe wR, 
WODOPFPWrROCO 








15.58 15.33 
16.40 16.13 





110° 





© OO =F Od Sd OT C9 C9 0 
~ ot 


DOwUc few 





——< 


120° 





OO OD FO OD OVP 09 09 09 
DOWwWUnowoens 
RSESSARASE 


OTTO 





TABLE VIII.—MIDDLE ORDINATES FOR RAILS 25 


LENGTH OF RaiIL-CHORD. 


























D D 
60 45 33 32 30 28 26 | 24 | 22 | 20 | 18 | 16 
1° | .078 | .048 | .025 | .022 | .020} .017 | .015 |.013).011}.009).007).006} 1° 
2 157 | .088 | .051 | .045 | .039 | .084 | .080 | .025;.021/.017;.014).011| 2 
3 235 | .182 | .069 | .067 | .059 | .051 | .044 | .0388) .032}.026).021].017| 3 
4 314 | .176 | .090°| .089 | .079 | .068 | .059 | .050) .042).035).028}.022} 4 
5 391 | .221,| .118 | .112 | .098 | .086 | .074 |.063) .053) .044| .035}.028} 5 
6 471 | .264 143 134 | .118 | .103 | .088 |.075) .063) .052| .042|.034} 6 
7 550 | .809 | .166 | .156 | .137 | .120 | .103 | .088) .074/.061|.049].039| 7 
8 620 | .353 190 | .179 | .157 | .1387 | .118 |.100).084).070/.057|.045) 8 
9 706 | .897 | .214.| .201 | .177 | .154 | .183 |.113) .095) 078] .064/.050| 9 
10 785 | .441 237 | .223 | .196 | .171 | .147 |.126). 105) .087|.071).056| 10 
ll 863 | .485 | .261 245 | .216 | .188 | .162 |.138/.116|.096).078].061| 11 
12 942 | .529 | .285 | .268 | .235 | .205 | .177 |.151).127).105) .085) .067| 12 
14 {1.098 | .617 | .3882 | .3812 | .274 | .238 | .206 |.175|.147|.122|.099).078| 14 
16 |1.255 ; .705 | .3879 | .856 | .313 | .273 | .235 |.200).168/.189).113'.089| 16 
18 (1.411 | .793 | .426 | .400 | .352 | .3807 | .264 |.225}.189).156|.127:.100| 18 
20 {1.567 | ~880 | .473 | .445 | .3891 | .3840 | .293 |.250).210).174;.141].111| 20 
24 |1.878 |1.055 | .567 | .531 | .467 | .407 | .3851 |.299).251). 207]. 168}. 133 oa 





40 (3.111 11.742 | .934 | .878 | .772 | .672 | .579 |.493].414|.342].277|.219| 40 
45 |3.491 |1.952 11.046 | .983 | .863 | .752 | .648 |.552}. 463] .383].305|.245| 45 
50 13.867 (2.159 |1:156 |1.087 | .955 | .831 | .716 |.610|.512) .423|.343].271| 50 























TABLE IX.— DIFFERENCE IN ELEVATION OF RAILS ON 
CURVES 


VELOCITY IN MILES PER Hour. 








10 15 20 25 30 35 40 45 50 55 | 60 | 65 





° 
oO 
fom] 
— 
ow 
j=) 
he) 
ow 
S 
we 
oO 
S 
Oo 
— 
So 
= 
So 
S> 
co 
— 
— 
— 
for) 
i 
es 
Cw 
— 
J 
iw) 
a 
oS 
oO 
CAS) 
— 
— 

















_ 
a 
([=} 
oe) 
fa) 
— 
ce 
oO 
w 
— 
co 
cs 
co 
~J 
3 
—_ 
— 
co 
or 
Yoo] 





26 TABLE X.—INCHES IN DECIMALS OF A FOOT 








| 
iat 1-0 | -1 | 2°) 58 aero om aye A, | Olea ene 








0 Foot} .0833) 1667) .2500) .3333°.4167/ .5000 .5833 6667) .7500 | .8333).9167) 0 


1-82 | .0026! .0859' 1693] 2526) .3359) 4193] 5026) .5859 .6693) .7526).8359) .9193] 1-32 
1-16 | .0052) .0885) .1719} .2552) .38885) .4219) 5052) 5885, 6719} .7552) .83885) 9219) 1-16 


3-32 | .0078 O11) .1745) 2578 3411 |.4245).5078) 5911/6745). 7578 811.9245) 3-32 

: 38. .6771| .7604| .8438| 9271] 1-8 
0130) .0964) .1797| 2630) 8464) 4299) .5130) 5964! .6797|.7630| 8464) - 
3-16 |.0156) .0990) 1823] .2656| 8490, .4323) .5156| .5990| .6823|.7656| 8490-9323] 3-16 
7-32 |.0182| 1016! 1849] 2682) 3516! .4349| .5182| .6016] .6849] .7682 .8516) .9349| 7-32 


1-4 |.0208) 1042) 1875) .2708] .3542) 4375) .5208) .6042/ .6875! 7708) .8542) .9375| 1-4 
9-32 | .0284| . 1068) 1901] .2734| 8568) .4401).5284) .6068) .6901|.7734) .8568) .9401} 9-32 
5-16 | .0260) .1094) 1927) 2760] 8594) .4427/ 5260! .6094! 76927) 7760) .8594! .9427) 5-16 
11-82 | .0286) .1120) 1953] .2786} .3620) .4453) .5286) .6120| .6953) .7786| .8620) .9453)11-32 
3-8 | .0313) .1146) 1979] .2813) 8646) 4479 .5313) 6146) 6979) .7813) 8646 9479) 3-8 

0339) . : : 4 .6172) .7005) . 7839} .8672! 9505) 13-32 
7-16 | .0365] 1198) 2031] .2865| 3698) |4531|.5365) |6198) . 7031 . 7865) .8698) 9531) 7-16 
15-82 |.0891) .1224) 2057} .2891| 38724] .4557| 5391) .6224| .7057) .7891| 8724) .9557| 15-32 


1-2 |.0417| .1250) 2083] 2917] 8750) .4583} 5417) .6250).'7083).7917) 8750) .9583} 1-2 
17-82 | .0443) 1276] 2109] .2943) 3776] 4609) .5443] .6276] 7109) .'7943) 8776) 9609] 17-32 
9-16 |.0469| .1302) .2135| 2969) .8802! .4635] .5469) . 6302) 7135] 7969] 8802] .9635| 9-16 
19-82 |.0495| 1328) 2161] 2995] 8828) .4661/.5495) 6328) .’7161| .7995| 8828 .9661]19-32 
5-8 |.0521|. 1354) 2188) 3021] 8854) 4688 5521) .6354/.7188) .8021] .8854) 9688] 5-8 
.0547| .1380| 2214] .3047| .3880 6380] .7214 .8047| 8880) .9714| 21-32 

11-16 | .0573|.1406 |2240) 3073/3906) 4740 5573 6406) .7240| 8073, .8906) .9740| 11-16 
5 32) 7266) .8099. 8932] 9766] 23-32 

3-4 |.0625) .1458) .2299| .3125] .3958) .4792 5625 6458] .'7292/ .8125! .8958| .9792] 3-4 
0651). 3151] .398 5651) .6484) 7318) .8151) .8984) . 9818] 25-32 
13-16 |.0677) 1510, .2344| |3177| .4010 .4844) .5677/ 6510) .7344| |8177| .9010) .9844|13-16 
27-32 | .0703 . 1536] .2370| 8203) .4036| .4870| .5703, .6536| .7370| 8203) .9036| .9870/27-32 
7-8 | .0729| 1563) .2396| 8229] 4063 4896) 5729) . 6563] . 7396] 8229] 9063) .9896| 7-8 
29-32 | .0755) 1589, 2422] 3255] .4089| 4922) 5755) 6589) 7422) .8255) .9089 9922] 29-32 
15-16 | 0781) .1615! 2448) .3281|.4115) .4948) 5781) .6615| .7448| .8281/ .9115) 9948] 15-16 
81-82 | 0807) . 1641) 2474) -3307'| 4141] 4974) .5807| .6641| 7474) .8307|.9141| 9974] 31-39 


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0 1 2 3 4 5 6 ¢ 8 9-| 10.) 11 








TABLE XI.—-METRIC CURVES 


Radius 
in Meters. 


3437.75 





Log. of 
Radius. 


3.536274 
3.235246 
3.059158 
2.934224 
2.837319 
2.758145 
2.691206 
2. 633223 
2.982081 
2.936335 
2.494955 


2.457181 
2.422434 
2.390266 
2.360320 


» 2.832811 


2.306002 
2.281200 
2.207741 
2.285489 
2.214325 
2.194148 
2.174870 


2.156416 
2.188717 
2.121715 
2.105357 
2.089596 
2.074391 
2.059704 
2.045501 
2.031751 
2.018427 
2.005503 
1.992956 


1.980765 
1.968911 
1.957375 
1.946141 
1.985194 
1.924520 
1.914105 
1.903938 
1.894008 
1.884302 
1.874813 
1.865530 


1.856445 
1.847549 
1.838836 
1.829298 
1.821928 
1.813720 
1.805668 
1.797766 


“1.790008 


1.782391 
1.774908 
1.767556 
1.760330 














Tang. 
Off. 








Degree of 
Equiv. 
U.S. 


Curve. 


0° 30’ 


WOO OONWIDD COOP WWD 
Cw 
oO 








27 





Defi. 
Angle 
20 m. 
Chord. 








10 





TABLE XII.—ACRES FOR RIGHT OF WAY 100 FT. WIDE 








11.501 


11.731 
11.961 








11.524 


11.754 
11.983 





11.547 


TT 
12.006 





11.570 


11.800 
12.029 





4.477 
4.706 


4.936 
5.165 
5.395 
5.624 
5. 854 
6.084 
6.313 
6.543 
6.772 
7.002 


7.281 
7.461 
7.691 
7.920 
8.150 
8.379 
8.609 
8.838 
9.068 
9.298 


9.527 
9.757 


. 9.986 


10.216 
10.445 
10.675 
10.905 
11.134 
11.364 
11.593 


11.823 
12.052 














11.616 
11.846 





12.075 


11.639 


11.869 
12.098 





11.662 


11.892 
12.121 





11.455 


11.685 


11.915 
12.144 








on 
2 


_ 


10 




































































TABLE XIII.—VELOCITY HEADS FOR VARIOUS SPEEDS 29 
VeLocity Heaps In FEET. 
Speed 
in Miles 
per Hour, 0 1 2 3 4 5 6 7 8 9 
0 0.00) 0.00) 0.00} 0.00} 0.01; 0.01] 0.01} 0.02} 0.02] 0.03 
1 0.04 9.04) 0.05) 0.06) 0.07) 0.08} 0.09} 0.10) 0.12)- 0.13 
2 OF4 0.16) C217) 0819) 0520% 0.221) 0.24) 0.26) 0.281 *0280 
3 0.32} 0.34) 0.386) 0.39) 0.41) 0.43) 0.46] 0.49] 0.51) 0.54 
4 0.57) 0.60} 0.63} 0.66) 0.69} 0.72) 0.75) 0.78] 0.821 0.85 
5 0.89; 0.92} 0.96} 1.00) 1.04) 1.07; 1.11) 1.15] 1.19} 1:24 
6 PeCSe teoel = beooy 1 Alieet-45) FVS50ie W551) 1-59) E64: 2 1e69 
Ce 174) ee 19) PSSay 1. SO 94) 22-0010 2-05) 22. LO 216) 22722 
8 Beato l| akoo| Choon ie c40l tee DO ReeDOle “2.60 le cnO9 ne cal) | moneL 
9 BESS eos) oLOUI vosOt oele) secQie o.oo odie oo 4L voces 
10 ee. Oo as0e| |orO9 onthe orcs! vacQlie o.09 |. 4.06 an4e 14 4522 
ll 4.30) 4.37) 4.45) 4.58) 4.61) 4.69] 4.78] 4.86] 4.94) 5.03 
12 Hele) Ghiec0)  Dacsle DO. olen os46) S5E551) 52641 5.731 5.82 vp Ok 
13 6.00! 6.09} 6.19} 6.28) 6.37) 6.47) 6.57) 6.66] 6.76). 6.86 
_ 14 629616 7206)  ek6l, 26S i200) CAG bie Gil Weis wi css 
15 7.99} 8.09} 8.20) 8.31) 8.42) 8.53) 8.64) 8.75) 8.86] 8.97 
16 *9.09} 9.20) 9.82} 9.43) 9.55) 9.66) 9.78) 9.89] 10.02) 10.14 
17 10.26} 10.38) 10.50} 10.62} 10.75) 10.87) 11.00) 11.12] 11.25) 11.37 
18 11.50). 11.63} 11.76} 11.89) 12.02) 12.15) 12.28) 12.41) 12.55) 12.68 
19 12.82) 12.95! 13.09} 18.22) 18.36] 18.50) 13.64] 18.78] 13.92) 14.06 
20 14.20) 14.34) 14:49) 14.63) 14.77) 14.92) 15.06] 15.21] 15.36) 15.51 
21 15.66) 15.80) 15.96} 16.11} 16.26) 16.41] 16.56) 16.72) 16.87] 17.03 
22 17.18} 17.34] 17.50) 17.65) 17.81) 17.97) 18.13] 18.29] 18.45) 18.62 
23 18.78) 18.94) 19.11] 19.27) 19.44} 19.60) 19.77) 19.94! 20.11] 20.28 
24 20.45] 20.62) 20.79) 20.96) 21.14) 21.31) 21.48) 21.66| 21.88) 22.01 
20 22.19) 22.37) 22.54) 22.72) 22.90) 28.08) 23.27| 23.45) 23.63] 23.81 
26 24.00} 24.18) 24.37) 24.55) 24.74) 24.93] 24.12] 25.31] 25.50) 25.69 
27 25.88| 26.07] 26.26) 26.46] 26.65| 26.85] 27.04] 27.24] 27.44] 27.63 
28 27.83| 28.03] 28.23) 28.43) 28.63) 28.83} 29.04) 29.24) 29.45| 29.65 
29 29.86} 30.06) 30.27) 30.48) 30.68] 30.89) 31.10] 31.381] 31.53] 31.74 
30 31.95! 32.16) 32.38] 32.59) 32.81) 33.02] 33.24] 33.46] 33.68) 33.90 
al 34.12 34.34! 34.56) 34.78) 35.00) 35.22) 35.45! 35.67) 35.90] 36.13 
32 36.35) 36.58] 36.81] 37.04] 37.27) 37.50! 37.73] 37:96) 38.19| 38.43 
33 38.66) 38.89) 39.13) 39.37) 39.60) 39.84) 40.08) 40.32) 40.56} 40.80 
34 41.04} 41.28) 41.52) 41.77) 42.01; 42.25) 42.50) 42.75| 42.99) 43.24 
35 43.49] 48.74] 43.99) 44.24) 44.49) 44.74] 44.99} 45.24] 45.50) 45.75 
36 46.01) 46.26) 46.52) 46.78) 47.04) 47.29) 47.55] 47.81) 48.08] 48.34 
an 48.60) 48.86) 49.13} 49.39) 49.66; 49.92] 50.19] 50.46) 50.72) 50.99 
38 51.26) 51.53) 51.80) 52.07) 52.35) 52.62) 52.89) 58.17) 538. 44] 53.72 
39 54.00} 54.27) 54.55) 54.83) 55.11) 55.39) 55.67) 55.95) 56.23) 56.52 
40 56.80) 57.08) 57.37] 57.66) 57.94) 58.23) 58 52] 58 81] 59 09] 59.38 
41 59.68} 59.97) 60.26} 60.55) 60.85} 61.14] 61.438] 61.73) 62.03] 62.32 
42 62.62) 62.92) 68.22) 68.52) 63.82) 64.12) 64.42] 64.73] 65.03) 65.33 
43 65.64). 65.95) 66.25! 66.56) 66.87) 67.17] 67.48] 67 79| 68.10) 68.42 
44 68.73) 69.04) 69.35) 69.67) 69.98] 70.30) 70.62) 70.93) 71.25) 71.57 
45 71.89) 72.21) 72.53) 72.85) 73.17) 73.49) 73.82) 74.14) 74.47) 74.79 
46 75.12) 75.44| 75.77| 76.10) 76.43] 76.76| 77.09) 77.42) 77.75| 78.09 
47 78-42) 78.75| 79.09) 79.42) 79.76) 80.10) 80.48) 80.77) 81.11) 81.45 
48 81.79! 82.13) 82.48] 82.82) 88.16) 83.50] 83.85} 84.19) 84.54) 84.89 
49 85.24| 85.58] 85.93) 86.28] 86.63) 86.98] 87.34] 87.69) 88.04) 88.40 
0.0 1.0 | 2.00% 3.00N5 4.0. 15:0 6.0 7.0 8.0 9.0 

50 88.75) 92934} 95.99) 99.721103.52/107.39 }111.33/115.34 |119. 42 |123.58 
60 127 .80}132. 10/136. 46)140.90)145.41}145.99 |154. 64/159. 36 |164.15/1€9.02 
70 173. 95/178 96) 184.03) 189.18}194. 40/199 .69 |205.05 1210.48 |215.98 {221.56 
80 297 20/232. 92|238.70|244.56/250. 49/256. 49 | 262.56 |268.70 |274.91 |281.20 
90 300.47 334 02/340.94 |847.94 


pt BS 293.98 





307 .04)313.68 320.39 1327.17 





30 


TABLE XIV.—GRADES AND GRADE ANGLES 























Feet 
Feet per | Inclina- |} per | Feet per | Inclina- 

ile. tion. Sta-| Mile. tion. 

tion. 
(2) Date O- a8 
.528 21 51 |. 26.928 17 32 
1.056 41 .52 | 27.456 17 53 
1.584 1 02 .538 | 27.984 18 13 
2.112 1 23 .54 | 28.512 18 34 
2.640 1 43 .55 | 29.040 18 54 
3.168 2 04 .56 | 29.568 19 15 
3.696 2 24 .57 | 30.096 19 36 
4,224 2 45 .58 | 380.624 19 56 
4.752 8 06 .59 | 381.152 20 17 
5.280 83 26 .60 | 31.680 20 38 
5.808 8 47 .61 | 382.208 20 58 
6.336 4 08 .62 | 382.736 21 19 
6.864 4 28 .63 | 383.264 21 39 
7.392 4 49 .64 | 383.792 22 00 
7.920 5 09 .65 | 84.320 22. 21 
8.448 5 30 .66 | 384.848 22 41 
8.976 5 51 .67 | 35.376 23 02 
9.504 6 11 .68 | 35.904 23 23 
10.032 6 382 .69 | 386.432 23 43 
10.560 6 53 -70 | 386.960 24 04 
11.088 7 13 TL | 37.488 24 24 
11.616 Y 34 .72 | 388.016 24 45 
12.144 7 54 73 | 38.544 25 06 
12.672 8 15 74 | 39.072 25 26 
13.200 |. 8 36 75 | 39.600 25 47 
13.728 8 56 76 | 40.128 26 08 
14.256 917 77 | 40.656 26 28 
14.784 9 38 Sti 41.184 26 49 
15.312 9 58 79 | 41.712 27 09 
15,840 10 19 .80 | 42.240 27 30 
16.368 10 39 .81 | 42.768 27 51 
16.896 11 00 .82 | 43.296 28 11 
17.424 11 21 .83 | 43,824 28 32 
17.952 11 41 .84 | 44.352 28 53 
18.480 12 02 .85 | 44.880 29 13 
19.008 12 23 .86 | 45.408 29 34 
19.536 12 43 .87 | 45.936 29 54 
20.064 13 04 .88 | 46.464 30 15 
20.592 13 24 .89 | 46.992 30 36 
21.120 13 45 .90 | 47.520 80 57 
21.648 14 06 91 | 48.048 31 17 
22.176 14 26 .92 | 48.576 31 38 
22.7 14 47 .93 | 49.104 31 58 
23.232 15 08 .94 | 49.632 32 19 
23.760 15 28 .95 50.160 82 39 
24.288 15 49 .96 | 50.688 33 00 
24.816 16 09 .9¢ | 51.216 33 21 
25.344 16 30 .98 | 51.744 33 41 
25.872 16 51 .99 | 52.272 34 02 
26.400 17 11 | 1.00 | 52.800 34 23 























Feet 
per | Feet per |Inclin- 
Sta-|} Mile. | ation. 

tion. 
ot. Via 
1.01 | 58.3828 34 43 
1.02 | 53.856 35 04 
| 1.08 | 54.384 35 24 
1.04 | 54.912 35 45 
1.05 | 55.440 36 05 
*1.06 55.968 36 26 
1.07 | 56.496 36 47 
1.08 | 57.024 37 08 
1.09 % 552 37 28 
1.10 | 58.080 37 49 
1.11 | 58.608 38 09 
Lee 59.136 88 30 
1.13 59.664 88 51 
1,14} 60.192 39 11 
1.15 60.720 39 32 
1.16 61.248 39 53 
dime atte 61.776 40 13 
1.18 | 62.304 40 34 
1.19 62.832 40 54 
1.20 | 63.360 41 15 
1.21 | 63.888 | 41 35 
1.22 | 64.416 | 41.56 
1.23 64.944 42 17 
1,24 65.472 42 38 
1.25 66.000 42 58 
1.26 66.528 43 19 
deel 67.056 43 39 
1.28 | 67.584 44 00 

1.29 68.112 44 21c 

1.380 | 68.640 44 41 
1.31 | 69.168 45 02 
1.32 | 69.696 45 23 
1:33 | > 70.224 45 43 
1.34 70.752 46 04 
1.35 | 71.280 46 24 
1.36} 71.808 46 45 
1.37 | 72.386 47 06 
1.38} 72.864 47 26 
1.39 | 73.392 47 47 
1.40 73.920 48 08 
1.41.| 74.448 48 28 
1.42 | 74.976 48 49 
1.43 75.504 49 09 
|| 144] 76.0382 49 30 
1345 76.560 |~+49 51 
1.46 77.088 50 11 
1.47 77.616 50 52 
1.48 | 78.144 50 &2 
1.49 78.672 51 13 
1.50 | 79.200 51 34 


TABLE XIV.—GRADES AND GRADE ANGLES ol 






































Feet Feet Feet 

per |Feet per} Inclina-|} per | Feet per} Inclina-|| per | Feet per | Inclina- 
Sta- | Mile. tion. Sta- Mile. tion. Sta- Mile. tion. 
tion. tion. tion. 

L51| 79.728 51 54 || 2.05 | 108.240 | 1 10 28 || 5.10 | 269.280 | 2 55 10 
1.52 | 80.256 52 15 || 2.10 | 110.880 | 1 12 11 || 5.20 | 274.560 | 2 58 36 
1.53 | 80.784 |. 52 36 || 2.15 | 118.520 | 1 13 54 || 5.380 | 279.840 | 3 02 09 
1.54 | 81.312 52 56 || 2.20 | 116.160 | 1 15 87 || 5.40 | 285.120 | 3 05 27 
1.55 | 81.840 53 17 || 2.25 | 118.800 | 1 17 20 || 5.50 | 290.400 | 3 08 53 
1.56 | 82.368 53 37 {| 2.80 | 121.440 | 1 19 03 || 5.60 | 295.680 | 3 12 19 
1.57 | 82.896 53 58 || 2.385 | 124.080 | 1 20 46 || 5.70 | 300.960 | 3 15 44 
1.58 | 83.424 54 19 || 2.40 | 126.720 | 1 22 29 || 5.80 | 306.240 | 3 19 10 
1.59 | 83.952 54.39 || 2.45 | 129.860 | 1 24 12 || 5.90 | 311.520 | 3 22 36 
1.60 | 84.480 55 00 || 2.50 | 182.000 | 1 25 56 |} 6.00 | 316.800 | 3 26 OL 
1.61; 85.008 55 21 || 2.55 |. 134.640 | 1 27 39 || 6.10 | 322.080 | 3 29 27 
1.62 | 85.586 55 41 || 2.60 | 187.280 | 1 29 22 |} 6.20 | 327.360 | 3 32 52 
1.63 | 86.064 56 02 || 2.65 | 139.920 | 1 31 05 || 6.30 | 332.640 | 3 36 18 
1.64 | 86.592 56 22 || 2.70 | 142.560 | 1 32 48 || 6.40 | 387.920 | 3 39 43 
1.65} 87:120|- 56 43 || 2.75 | 145.200 | 1 34 31 || 6.50 | 348.200 | 3 43 08 
1.66 | 87.648 57 04 || 2.80 | 147.840 | 1 386 14 || 6.60 | 348.480 | 3 46 34 
1.67 | 88.176 57 24 || 2.85 | 150.480 | 1 387 57 || 6.70 | 353.760 | 3 49 59 
1.68 | 88.704 57 45 || 2.90 | 153.120 | 1 89 40 || 6.80 | 359.040 | 3 53 24 
1.69 | 89.232 58 06 || 2.95 | 155.760 | 1 41 23 || 6.90 | 364.320 | 3 56 50 
1.70 | 89.760 58 26 || 3.00 | 158.400 | 1 43 06 || 7.00 | 369.600 | 4 00 15 
1.71 | 90.288 58 47 || 3.05 | 161.040 | 1 44 49 || 7.10 | 374.880 | 4 03 40 
i.72 | 90.816 59 O07 || 3.10 | 163.680 | 1 46 32 || 7.20 | 380.160 | 4 07 06 
1.73 | 91.344 59 28 |) 3.15 | 166.3820 | 1 48 15 |} 7.30 | 385.440 | 4 10 31 
1.74 | 91.872 59 49 || 3.20 | 168.960 | 1 49 58 || 7.40 |} 890.720 | 4 13 56 
1.75 | 92.400 | 1 00 09 || 3.25 | 171.600 | 1 51 41 |) 7.50 | 896.000 | 4 17 21 
1.76 | 92.928 | 1 00 30 || 3.30 | 174.240 | 1 58 24 || 7.60 | 401.280 | 4 20 46 
1.77 | 93.456 | 1 00 51 || 3.85 | 176.880 | 1 55 O7 || 7.70 | 406.560 | 4 24 11 
1.78 | 93.984 | 1 01 11 |} 3.40 | 179.520 | 1 56 50 |] 7.80 | 411.840 | 4 27 36 
1.79 | 94.512 | 1 01 32 || 8.45 | 182.160 | 1 58 33 || 7.90 | 417.120 | 4 31 Ol 
1.80 | 95.040 | 1 01 52 || 3.50 | 184.800 | 2 00 16 |; 8.00 | 422.400 | 4 34 26 
1.81 | 95.568 | 1 02 13 || 3.55 | 187.440 | 2 01 59 |} 8.10 | 427.680 | 4 387 51 
1.82 | 96.096 | 1 02 34 || 8.60 | 190.080 | 2 03 42 || 8.20 | 482.960 | 4 41 16 
1.83 | 96.624 | 1 02 54 || 3.65 |} 192.720 | 2 05 25 || 8.80 | 488.240 | 4 44 41 
1.84; 97.152 | 1 03 15 || 3.70 | 195.360 | 2 07 08 || 8.40 | 443.520 | 4 48 06 
1.85 | 97.680 | 1 03 35 |} 3.75 | 198.000 | 2 08 51 || 8.50 | 448.800 | 4 51 30 
1.86 | 98.208 | 1 03 56 || 3.80 | 200.640 | 2 10 34 ate 454.080 | 4 54 55 
1.87 | 98.7386 | 1 04 17 || 3.85 | 203.280 | 21217 || 8.70 | 459.360 | 4 58 20 
1.88 | 99.264 | 1 04 87 || 3.90 | 205.920 | 2 14 00 || 8.80 | 464.640 | 5 O1 44 
1.89 | 99 792 | 1 04 58 || 3.95 | 208.560 | 2 15 43 || 8.90 | 469.920 | 5 05 10 
1.90 | 100.3820 | 1 05 19 {| 4.00 | 211.200 | 2.17 26 || 9.00 | 475.200 | 5 08 34 
1.91 | 100.848 | 1 05 39 || 4.10 | 216.480 | 2 20 52 || 9.10 | 480.480 | 5 11 59 
1.92 | 101.376 | 1 06 00 || 4.20 | 221.760 | 2 24 18 || 9.20 | 485.760 | 5 15 23 
1.93 | 101.904 | 1 06 20 || 4.80 | 227.040 | 2 27 44 || 9.80 | 491.040 | 5 18 48 
1.94 | 102.432 | 1 06 41 |] 4.40 | 232.320 | 2 81 10 |} 9.40 | 496.320 | 5 22 12 
1.95 | 102.960 | 1 07 02 || 4.50 | 287.600 | 2 34 36 || 9.50 | 501.600 | 5 25 37 
1.96 | 103.488 | 1 O07 22 || 4.60 | 242.880 | 2 88 O1 || 9.60 | 506.880 | 5 29 OL 
1.97 | 104.016 | 1 07 43 || 4.70 | 248.160 | 2 41 27 || 9.70 | 512.160 | 5 32 25 
1.98 | 104.544 | 1 08 04 |} 4.80 | 258.440 | 2 44 53. || 9.80} 517.440 | 5 35 50 
1.99 | 105.072 | 1 08 24 | 4.90 | 258:720 | 2 48 19 || 9.90 | 522.720 | 5 39 14 
2.00 | 105.600 | 1 08 45 || 5.00 | 264.000 | 2 51 45 |/10.00 | 528.000 | 5 42 38 





20°. 


21°. 


S OCOMDVHMARWHHDS COMHORwWiIUS DONIOoA wie 


iw) 
re) 
° 


23°. 


24° 


PWOIRMRWIDHS DHYRMAwWNWES SHOT wi L 


. . 
C, = 0, 
y 
. ° 


TABLE XV.—FOR OBTAINING BAROMETRIC 


0.00 


16832 
16970 
17107 
17243 
17379 
17514 
17648 
17781 
17914 
18046 


18178 
18308 
18438 
18568 
18697 
18825 
18953 
19080 
19206 
19332 


19457 
19582 
19706 
19829 
19952 
20074 
20196 
20317 
20428 
20558 


20677 
20796 
20914 
21032 
21150 
21266 
21383 
21498 
21614 
21728 


21843 
21957 
22070 
22183 
22295 
22407 
22518 
22629 
22739 
22849 
22959 
23068 
23176 
28285 
23392 
23500 
23606 
23713 





23819: 


23924 





0.02 


0.04 





0.06 


18256 
18386 
18516 
18645 
1877 

18902 
19029 
19156 
19282 
19407 


19532 
19656 
19780 
19903 








PANAMA UNT TOHRTRRRRRA AAW 


Pt ittintitrwrwroio wri wd to~eD WOWWBDODARAD PAD 


89 2 IOIW IW IW WIW 


Sis 





HEIGHTS IN FEET 
Barom- 
eter. 
Inches 0.00 0.02 0.04 0.06 
—w 
25°.0 24029 24050 24071 24092 
el 24134 24155 24176 24197 
2 24238 24259 24280 24301 
3 24342 24363 24384 24404 
4 24446 24466 24487 24508 
5 24549 24569 24590 24610 
6 24651 24672 24692 24713 
vi 247 DAGT4 24794 24815 
8 24855 24876 24896 24916 
9 24957 2497 24997 25018 
26°.0 25058 2507 25098 25118 
* 25159 25179 25199 25219 
2 25259 2527 25299 25319 
3 25359 25379 25399 25419 
4 25458 25478 25498 25518 
5 25557 95577 25597 25617 
.6 25656 25676 25696 25715 
v 25755 25774 25794 25813 
8 25853 25872 25892 25911 
9 25950 2597 25989 26009 
2'7°.0 26048 26067 26086 26106 
1 26145. 26164 26183 26203 
2 26241 26260 26280 26299 
3 26337 26357 2637 26395 
4 26433 26452 26472 26491 
5 26529 26548 26567 26586 
6 26624 26643 26662 26681 
4 26719 26738 26757 26776 
8 26813 26832 26851 26870 
pee to 26908 26926 26945 26964 
28°.0 27001 27020 27039 27058 
1 27095 27114 27132 27151 
2 27188 27207 27225 27244 
3 27281 27299 27318 27336 
4 27373 27392 27410 27429 
5 27466 27484 27502 27521 
6 a 557 2757 27594 27612 
v; 27649 27667 27685 27704 
8 27740 27758 Q7777 27795 
9 27831 27849 27867 27885 
29°.0 27922 27940 27958 2797 
a 28012 28030 28048 28066 
2 28102 28120 28138 28156 
3 28192 28209 28227 28245 
4 28281 28299 28317 28334 
5 28370 28388 28405 28423 
6 28459 28476 28494 28512 
a2 28547 28565 28582 28600 
8 28635 28653 2867 28688 
9 28723 28741 28758 28776 
30°.0 28811 28828 28846 28863 
a 28898 28915 28933 28950 
2 28985 29002 29020 29037 
3 29072 29089 29106 29124 
4 29158 29175 29192 29210 
5 29244 29261 2927; 29296 
6 29330 29347 29364 29381 
af 29416 29433 29450 29467 
8 29501 29518 29535 29552 
9 | 29586 29603 £9620 29637 





26125 
26222 
26318 
26414 
26510 
26605 
26700 
26795 
26889 
26983 
27076 
27169 
27262 
27355 
27447 
27539 
27631 
20722 
27813 
27904 


Padi besh tebe ot peek pal ead Ve ae ee lc gl dl Le ee a cee ae ee ee wWwwwwwwwww WW 0H WWW WWWD 


WEAEAIAT$TTAFIWDW WH BCG MMW WWWW WOMDHHODHWOOO DOODOBDOOOOO DOSOOSOOOSOO SOSOBHHEHEEHE 





34 TABLE XVI,—COEFFICIENT OF CORRECTION FOR 




































































TEMPERATURE 
| | 
t+ t/ — 64° t+t — 649}! t+v — 64°! t-+t/ — 64° 
fe / / / 
tate 900 [b+ Oh o00 bE Gop ert 900 

pe PSS a i ON es 
20° .0489 65° | + .0011 || 110°] + .0511 |) 155° 1011 
21 | — .047 66 “ooze L411 (0522 | 156 1022 
22 0467 - || 67 0033 |) 112 (0533. |, 157 1033 
23 0456 68 .0044 || 113 0544 |, 158 1044 
24 0444 69 [0056 |} 114 0536 | 159 1056 
25 0433 70 0067 |] 115 0567 | 160 1067 
26 0422 “1 ‘0078 || 116 0578 | 161 107 
2Q7 0411 "2 -0089 || 117 0589 | 162 1089 
28 0400 "3 ‘0100 || 118 0600 |) 163 1100 
29 0389 74 ‘0111 || 119 0611 | 164 1111 
30 0378 (Bs 10122 || 120 | + .0622 | 165 1122 
31 | — .0367 "6 | + .0133 || 121 | * .0683 || 166 | + .1133 
32 0356 iG (0144 || 122 0644 || 167 1144 
33 0344 "8 (0156 || 128 0656 || 168 ‘1156 
34 0333 "9 .0167 || 124 0667 || 169 1167 
35 0322 80 (0178 || 125 0678 || 170 1178 
36 0311 81 :0189 || 12 0689 |! 171 "1189 
37 0300 82 :0200 || 127 0700 || 172 “1200 
38 0289 83 (0211 || 128 O711 || 178 ‘1211 
39 027 84 10222 || 129 0722 || 174 | 1928 
40 0267 85 '0233 || 130 | + 10733 |! 175 £1233 
41 | — .0256 g6 | + .0244 || 131 0744 || 176 | + 11244 
42 0244 87 (0256 || 132 156 || 177 1256 
43 0233 88 (0267 || 133 or67 |) 178 1267 
44 0222 89 (0278 |) 134 or78 || 179 1278 
45 0211 90 (0289 |) 135 gg || 180 1289 
46 0200 91 ‘0300 || 136 0800 | 181° "1300 
? 0189 92 (0311 || 137 0811 | 182 '1311 
48 0178 93 (0322 || 138 0822 || 188 1322 
49 0167 94 :0333 || 139 0833 | 184 1333 
50 | — .0156 95 10344 |! 140 | + 10844 |) 185 1344 
51 0144 96 | + .0856 || 141 0856 | 186 -| + 11356 
52 0133 97 0367 || 142 0867 | 187 1367 
53 0122 98 :0878 || 148 0878 | 188 '1378 
54 0111 99 -0389 || 144 0889 || 189 "1389 
5B 0100 100 :0400 |} 145 0900 || 190 -1400 
56 0089 101 :0411 || 146 0911 | 191 ‘1411 
BY 007 102 (0422 || 147 0922 | 192 1422 
58 0067 103 :0433 || 148 0933 | 193 1433 
59 0056 || 104 10444 |} 149 0944 | 194 1444 
60 0044 || 105 (0456 || 150 | -+ .0956 | 195 "1456 
61 | — .0083 || 106 | + .0467 || 151 0967 | 196 | + 11467 
62 0022 || 107 (0478 || 152 0978 || 197 147 
63 0011 108 '0489 |) 153 0989 |) 198 1489 
64 0000 || 109 :0500 || 154 1000 |) 199 1500 














TABLE XVII.—CORRECTION FOR EARTH’ 


AND REFRACTION 


S CURVATURE 




















“we | He} Lo | we i} yo | ae |] we | Ae || we 
300 | .002 | 1300| .035 || 2300/ .108 || 3300) .223 || 4300 
400 | .003 | 1400 | {040 || 2400) “118 |] 8400) “287 || 4400 
500 | 


os 
or 
3 
S 
i= 
& 
2 
x 
3 
= 
a 
oo 0D 
SH 
oo 
= 
20 
ot 
Ss 
3 


























H° 





Miles 





SODMDRIOUP WOH 


— 











TABLE XVIII.—STADIA REDUCTIONS FOR READING 100 35 


DIFFERENCES IN ELEVATION 























Minutes. 0° 13 oe By 4° 5° 6° 72 8° | g° 
0 .00 | 1.74 | 3.49 | 5.23 | 6.96 8.68] 10.40} 12.10) 13.78) 15.45 
2 06) .1.80"). 3.55<f-5.28 1 7:02 8.74} 10.45) 12.15] 18.84] 15.51 
4 HU 1860. 3660s), 528402 707 8.80} 10.51) 12.21] 18.89} 15.56 
6 ‘7 \ 1.92) 3.66 | 5.40) 7:18 8.85} 10.57] 12.26] 18.95] 15.62 
8 Peo LUGSule Olle te OF460h TEED 8.91) 10.62] 12.32] 14.01] 15.67 
10 5 | 2:04) 3.78) 5762 1 7425 8.97} 10.68} 12.38] 14.06] 15.73 
12 .89 | 2.09 | 3.84 } 5:57 | 7.30 9.03) 10.74! 12.43] 14.121 15.78 
14 PAY) 26950-32900" 5:68) 7936 9.08} 10.79} 12.49] 14.17! 15.84 
16 TAVs) 2021 8195 oh 5269 17242 9.14) 10.85] 12.55] 14.23] 15.89 
18 52 | 2.27 4.01 | 5.75 | 7.48 9.20} 10.91] 12.60} 14.28] 15.95 
20 258) | 2.83 1, 4.0771 5.80 | 7:53 9.25| 10.96] 12.66} 14.34] 16.00 
22 .64 | 2.38 | 4.13 | 5.86 | 7.59 9.31} 11.02] 12.72] 14.40] 16.06 
24 PO) 244 4718 75092 be 7.65 9.37} 11.08) 12.77! 14.45] 16.11 
26 -76 | 2.50 | 4.24 | 5.98 | 7.71 9.43] 11.13) 12.83] 14.51] 16.17 
28 S81) 225674138091 6164) 7276 9.48} 11.19] 12.88] 14.56] 16.22 
30 .87 | 2.62 | 4.36 | 6.09 | 7.82 9.54) 11.25] 12.94! 14.62] 16.28 
32 Osa) ec67} 4042.51, 6.158) 7588 9.60} 11.30} 18.00} 14.67) 16.33 
3 299% 2°73 1 4.48-) 6:21 1-794 9.65! 11.36] 13.05) 14.73] 16.39 
36 1.05 | 2.79 | 4.538 | .6.27 | 7.99 9.71] 11.42] 18.11} 14.79] 16.44 
38 J].11 | 2.85 | 4.59 | 6.33 | 8.05 9.77) 11.47] 18.17) 14.84] 16.50 
40 1.16 2.91 | 4.65 | 6.38 | 8.11 9.83] 11.53) 18.22} 14.90} 16.55 
42 Leger 2097) ASELat 6044. | 8147 9.88} 11.59) 13.28} 14.95) 16.61 
44 1.28 | 3.02 | 4.76 | 6.50 | 8.22 9.94! 11.64] 13.33] 15.01] 16.66 
46 1.34 | 3.08 | 4.82 | 6.56 | 8.28 | 10.00} 11.70] 13.89] 15.06! 16.72 
48 1.40 | 3.14 | 4.88 | 6.61 | 8.34 | 10.05) 11.76] 13.45] 15.12] 16.77 
50 | 1.45 | 3.20 | 4.94 | 6.67 | 8.40 | 10.11; 11.81] 13.50) 15.17; 16.83 
52 1.51 | 3.26 | 4.99 | 6.73 | 8.45 | 10.17) 11.87] 18.56] 15.23) 16.88 
54 ee Belay: | 3.31 | 5.05 | 6.78 | 8.51 | 10.22} 11.938} 13.61] 15.28] 16.94 
56 1.63 | 3.37 | 5.11 | 6.84 | 8.57 | 10.28] 11.98) 138.67) 15.34] 16.99 
58 1.69 | 3.48 | 5.17 | 6.90 | 8.63 | 10.34] 12.04] 18.73) 15.40] 17.05 

42.960) 1.74) | 3.49 | 5.23 | 6.96 | 8.68 | 10.40) 12.10) 13.78] 15.45. 17.10 
AAs; 01 .02 03 .05 06 07 .08 Slit el 

1.00 01 03 .04 .06 08 .09 bl 15 .16 
1.25 02 | .03 mOp 08 .10 wig! 14 18 ted, 
fte 
































CORRECTIONS TO HORIZONTAL DISTANCES 

















Vertical ’ 

Angle. I 
Qe aS na ea 
l 03 04 
2 ote 14 
3 2 ok 
4 .49 53 
5 76 81 
6 1.09 31.15 
7 1.49 1.56 
8 1.94 2.02 
9 2.45 2.54 








20/ 30/ 40’ 50/ Vertical 
Angle. 
BEY 01 -O1 02 0° 
.05 .07 .08 10 1 
AY: 49 see a4 2 
384 Ob 41 45 3 
.5T 62 66 aval! 4 
86 92 .98 1.03 5 
aa 1.28 1.35 1.42 6 
1.63 1.70 1.78 1.86 7 
2.10 2.18 2.a0 2.36 8 
2.63 2:'t2 2.82 2.92 9 

















36 «TABLE XVIII—STADIA REDUCTIONS FOR READING 100 


DIFFERENCES IN ELEVATION 








Minutes. LO" | LTP) ISS BS ee ae 16° 2 |: Pe eee toe 
































17.59) 19.21) 20.81} 22.39] 23.93} 25.45) 26.94] 28.39) 29.81] 31.19 

20 17.65) 19.27; 20.87) 22.44] 238.99) 25.50} 26.99] 28.44] 29.86] 31.24 
22 17.70} 19.32] 20.92] 22.49) 24.04} 25.55} 27.04] 28.49] 29.90} 31.28 
24 17.76} 19.38] 20.97) 22.54) 24.09} 25.60} 27.09} 28.54] 29.95) 31.32 
26 17.81} 19.43) 21.03} 22.60) 24.14] 25.65| 27.13] 28.58] 30.00} 31.38 
28 17.86; 19.48) 21.08) 22.65) 24.19} 25.70} 27.18) 28.63] 30.04] 31.42 
30 17.92} 19.54) 21.13} 22.70) 24.24] 25.75] 27.23) 28.68] 30.09) 31.47 
32 17.97) 19.59) 21.18) 22.75} 24.29} 25.80) 27.28) 28.73] 30.14] 31.51 
34 18.03) 19.64) 21.24) 22.80) 24.34) 25.85] 27.33] 28.77] 30.19] 31.56 
36 18.08} 19.70) 21.29) 22.85) 24.39) 25.90] 27.388) 28.82) 30.23} 31.60 
38 18.14) 19.75) 21.34] 22.91) 24.44) 25.95) 27.43] 28.87] 30.28) 31.65 
40 18.19) 19.80) 21.39) 22.96} 24.49} 26.00] 27.48) 28.92] 30.382} 31.69 
42 18.24} 19.86) 21.45) 23.01) 24.55) 26.05| 27.52) 28.96] 30.37) 31.74 
44 18.30) 19.91) 21.50) 23.06) 24.60) 26.10) 27.57) 29.01} 30.41) 31.78 
46 18.35} 19.96) 21.55) 23.11) 24.65) 26.15| 27.62} 29.06] 36.46) 31.83 
48 18.41} 20.02) 21.60} 23.16) 24.70) 26.20} 27.67) 29.11) 30.51] 31.87 
50 18.46} 20.07) 21.66} 23.22) 24.75) 26.25) 27.72) 29.15). 30.55! 31.92 
52 18.51) 20.12) 21.71) 23.27) 24.80) 26.30) 27.77) 29.20] 30.60} 31.96 
54 18.57) 20.18) 21.76; 23.32) 24.85, 26.35) 27.81) 29.25] 30.65) 32.01 
56 18.62) 20.23) 21.81) 23.37) 24.90) 26.40) 27.86) 29.30} 30.69} 32.05 
58 18.68) 20.28) 21.87) 23.42) 24.95) 26.45) 27.91] 29.34] 30.74] 32.09 
60 18.73) 20.34) 21.92) 23.47) 25.00) 26.50) 27.96) 29.39] 30.78} 32.14 
15 14 15 16) 4,527) 2 19 20 21 23 24 25 
1.00 18 20 22; .ed| 20 27 28 30 32 33 
7 sig 23 25 27)... 429] ., Sel 34 36 38 40 42 

c 








CORRECTIONS TO HORIZONTAL DISTANCES 














Vertical , , ’ , , , Vertical 

"Angle: 0 10 20 30 40 50 ‘Arigtes 
10° 3.02 Bike Bee B.oe 3.43 3.53 10° 
ll 3.64 3.75 3.86 3.97 4.09 4.21 ll 
12 4.32 4.44 4.56 4.68 4.8] 4.93 12 
13 5.06 5.19 5.82 5.45 5.58 5.72 13 
14 5.85 5.99 6.13 6.27 6.41 6.55 14 
15 6.70 6.84 6.99 7.14 7.29 7.44 15 
16 7.60 7.75 7.91 8.07 8.23 8.39 16 
17 8.55 8.71 8.88 9.04 9.21 9.38 17 
18 9.55 9.72 9.89 10.07 10.24 10.42 18 
19 10.60 | 10.78 | 10.96 11.14 11.33 11.51 19 


TABLE XVIII.—STADIA REDUCTIONS FOR READING 100 37 


DIFFERENCES IN ELEVATION 





















































Minutes. 20cm ses | sees 24a ne oe 26° Pyke Nie 29° 
0 32.14} 38.46) 34.73} 35.97] 37.16) 38.380) 39.40} 40.45] 41.45] 42.40 
2 32.18] 33.50) 34.77] 36.01} 37.20) 38.34] 39.44] 40.49] 41.48) 42.43 
4 82.23! 383.54] 34.82! 36.05) 37.23! 38.38] 39.47! 40.52] 41.521 42.46 
6 32.27| 33.59] 34.86) 36.09) 37.27) 38.41) 39.51) 40.55] 41.55] 42.49 
8 32.32) 33.63] 34.90} 36.18) 37.31] 38.45) 39.54] 40.59) 41.58] 42.53 
10 82.36) 33.67] 34.94) 36.17) 37.35) 38.49} 39.58] 40.62) 41.61] 42.56 
12 32.41} 38.72] 34.98] 36.21) 37.39) 38.53) 39.61] 40.66] 41.65} 42.59 
14 32.45) 33.76} 35.02) 36.25) 87.43) 38.56} 39.65) 40.69] 41.68] 42.62 
16 32.49} 33.80) 35.07) 36.29) 387.47) 38.60] 39.69) 40.72) 41.71] 42.65 
18 32.54) 33.84] 35.11) 36.33) 37.51] 38.64] 39.72) 40.76| 41.74] 42.68 
20 32.58} 33.89) 35.15} 36.37) 37.54) 38.67| 39.76) 40.79) 41.77) 42.71 
22 32.63} 38.93} 35.19) 36.41) 37.58) 38.71] 39.79! 40.82) 41.81] 42.74 
24 32.67) 33.97) 35.23] 36.45) 37.62) 38.75) 39.83) 40.86] 41.84] 42.77 
26 82.72) 34.01) 35.27) 36.49} 37.66, 38.78| 39.86; 40.89} 41.87] 42.80 
28 32.76) 34.06] 35.31) 36.53] 37.70) 38.82) 39.90; 40.92) 41.90] 42.83 
30 32.80; 34.10} 35.36) 36.57) 37.74] 38.86) 39.93} 40.96) 41.93] 42.86 
32 32.85} 34.14) 35.40} 36.61) 37.77) 38.89] 59.97) 40.99} 41.97] 42.89 
34 32.89} 34.18) 35.44) 36.65) 37.81) 38.93] 40.00) 41.02] 42.00} 42.92 
36 32.93) 34.23] 35.48] 36.69) 37.85) 38.97] 40.04] 41.06] 42.03] 42.95 
38 32.98] 34.27] 35.52) 36.73} 37.89) 39.00) 40.07) 41.09] 42.06] 42.98 
40 33.02) 34.31] 35.56) 36.77) 37.93) 39.04] 40.11} 41.12} 42.09] 43.01 
42 33.07) 34.35] 35.60} 36.80) 37.96) 39.08} 40.14] 41.16] 42.12] 43.04 

ye 44 33.11) 34.40) 35.64) 86.84} 38.00) 39.11) 40.18] 41.19} 42.15] 43.07 
46 33.15} 34.44] 35.68! 36.88) 38.04) 39.15] 40.21] 41.22] 42.19} 43.10 
48 33.20) 34.48) 35.72] 36.92) 38.08) 39.18) 40.24] 41.26] 42.22] 43.13 
50 33.24) 34.52} 35.76] 36.96] 38.11] 39.22) 40.28) 41.29] 42.25] 43.16 
52 33.28) 34.57) 35.80] 37.00} 38.15) 39.26) 40.31) 41.32] 42.28) 43.18 
54 33.33) 34.61] 35.85] 37.04] 38.19} 39.29] 40.35) 41.35) 42.31] 43.2] 
56 33.37| 34.65} 35.89] 37.08] 38.23) 39.33) 40.38) 41.39] 42.34] 43.24 
58 33.41) 34.69) 35.93) 37.12) 38.26) 39.36; 40.42] 41.42) 42.37) 43.27 
60 33.46) 34.73] 35.97] 37.16) 38.30) 39.40) 40.45) 41.45) 42.40] 43.30 
15 26 Rew 29 30 381 Bee BR} 38D 36 BB YG 
1.00 13) Bilf .38 40 Al 43 .45 46 AT 48 
teZo 44 46 48 50 52 54 56 58 .60 62 

fte 
CORRECTIONS TO HORIZONTAL DISTANCES 

Vertical , , , Y , Vertical 

Abela 0 10 20 30 40) 50 Angle 
20° LUTON IS89-) 12307 12.26 12.46 12.65 20° 
21 12.84 | 13.04 | 13.23 13.48 13.63 13.83 21 
22 14.03 | 14.24] 14.44 14.64 14.85 15.06 22 
23 15NeF |) 15248") 1569 15.90 16.11 16.33 23 
24 16.54 | 16.76} 16.98 17.20 142 17.64 24 
25 17.86 | 18.08 | 18.31 18.53 18.76 18.99 25 
26 19.22 | 19.45 | 19.68 19.91 20.14 20.38 26 
27 20.61) 20.85 | 21.08 21232 21.56 21.80 27 
28 225044) 22:28" hmeoroe APA Ae 23.01 23.26 28 
29 23.50 | 238.75 | 24.00 24.25 24.50 24.75 29 





38 TABLE XIX.—MEAN REFRACTIONS IN DECLINATION 

















































































































































































































3| 3 DECLINATIONS. 
Todi cey ee ry 
| 34 | 
S| m | +20°) +15°| +10°|; +5° | +0° | —5° | —10°| —15°} —20° 
~ [Obey 18” | -12” | —07” | —02” 02” 07” 1e¢" 18” 23” 
@/|2 —18 | —12 | —07 | —02 02 07 12 18 23 
3 —17 | —11 | —06 | —Ol 03 08 13 19 25 
o | 4 —15 | —10 | —05 00 05 10 15 21 27 
aa | —10 | —05 00 | +05 10 15 20 26 32 
Oh | —15” | —10’| —05” 00” 05’ 10’ 15’ 20” ld 
2 —15 | —10 | —05 00 05 10 15 20 27 
& 3 —13 | —08 | —03_/ +02 yt eck | 17 23 29 
4 ~10 | —05 00 | +05 10 15 20 27 32 
5 —05 00 | +05 | +10 15 20 27 32 40 
_ | Oh | —13” | —08” | —02” 02” 08” 13” 18” 24" 29/’ 
o/2 = 1Bastie Ot. te — OL 03 09 14 19 25 31 
oo eS —10 | —05 00 05 10 15 20 26 32 
o 14 —05 00 | +05 10 15 20 26 32 3 
5 +07 | +12 | +17 23 29 36 43 51 | 1’ Ol 
0 10” | —05” 00” 05/” ad ee 20” 267 ao” 
ered =O 0B ube 08 Qn ioe 12 has aed] 22 28 34 
©|3 —05 00 03 08 13 19 25 31 38 
m4 00 | +05 10 15 20 26 32 39 4 
5 +15 | +20 | 26 32 39 46 56 | 1’ 06 | 1’ 19 
= iQ hel =-08/7 ly —02” 02”) 08” 13” Lil 2a 30" 36” 
3/2 | —-06 | 00 05 104s eae 20 26 32 39 
gees +02 | +07 12 17 23: 1ae OG 36 43 51 
a) 4 +04 | +09 14 20 25 31 8 
m5 +21 | +27 33 40 | 48 BTA OS ond” 23-7] att Al 
Oh | —05” 00” | 05” 10” 15” ay! OHNE A eiagart) Se Ai? 
ae —03 02 07 12 18 23 29 36 43 
1» | 8 +01 05 ll 16 22 28 34 4] 49 
wi 4 +08 12e) Sik 19 24-| 30 1 4 3 | 1’ 04 
5 +29 34 4] 49 597] 17 10 1" S4- eas eae 
S (8) h —()23"/ 02” 08’ | 13” 18% 94" 30” 36” 44” 
=| 2 00 05 0. feed 21 27 33 40 48 
es +02 10 Up my oc 27 33 40 48 57 
| 4 +13 18 23 29 35 jac Lat ewe vnb  k 
=| 5 +34 41 | 49 68 | 1710 | 1723 | 1’ 41 | 2 06 | 2 42 
0 h 00’’ 05”’ 107 | 57 Ds on 33” 40” hoe 
S42 03 07 13 18 24 30 36 44 52 
°o/|3 06 13 18 24 30 36 44 52 | 1’ 02 
Ria 17 22 28 35 42 L007 ee ee 
5 39 AT BT) A OFT. 20: 21" 37-30 o 00s koomeowe. 
S 0 h 02” 08’’ 13% Si Oar" 30” 36” 44”’ ROA 
aS 06 11 15 21 va 33 40 48 57 
tes ll 15 21 OT Ai 3 40 48 57 | 1/08 
m| 4 20 26 3 39 46 BEE} 1-07 Lf 1410 Ta oF 
1 5 45 63. | 08°) Mi6l W118i 17628) 2 B13 07-1 14228 
Oh 05" 10” 15” atv as 330) + a0! 48") 57” 
5 08 |; 14 19 25 31 38 46 |- 54 | 1’ 05 
19 | 3 12 18 24 30 37 | 44 53 04 | 1 18 
Rl 4 23 29 35 45 53!) 1/088) 118-401 131 Jeb 52 
5 49 59 | 1/10 | 124 | 152 | 2 07 | 2 44. | 8 46 5 48 
+20° | +15°| +10°} +5° | 0° —5° | —10°| —15° | —26° 





TABLE XIX.—MEAN REFRACTIONS IN DECLINATION 239 






























































































































































AR iM 
ers DECLINATIONS. 
= he | . seed 
3 | 3<| °o : ° / °o fo] | ° ° ° ° °o 
4) | +20°) 415°); +10°, +65 0° = —5° 10°) —15° | —20 
ejOh] 087) 13) 18” 24’) 30” 36” 44’) 52’ 1! 02" 
o|2 | Hi 16 | 22 28 SA* VAROAT 49 | 1’00 | 1 10 
1 3 TieTe 29 |7 «198 35 £21, 50 OY. 00° 151 11) Pes 
m4 aR0ia7 35 |>e 42 50 | 1’ 00 | 1/11 |:1 26 | 1 43 | 209 
1 5 B4 | 105-41’ 18 | VY 34 | 1-54 | 2 2418 11 | 4 38 | 815 
| | | 
Oh 10> 15’’| 2)” Py iad 33” 40”, 49” 57” 1 08” 
o | 2 14; 19 25 31 38 46 | 5411/05 | 1 18 
2 3 20 26 32:1 39 47 55 | 1/06 | 1 19 |- 1 36 
cae 32 39 46°) 2'52 (06 |-1719 71 35, | 57 |e 29 
5 1°00} 1/10 | 1’ 24 | 1°52 | 2 07 | 2 44 13 46 | 5 43 113° 06 
any @ Wetead 48 hay A824] B0 hin 86" bra 447th 52/01" 021g Bild? 
312 17 29 28 35 42 5O. |o1’ 00 4) 11.) 636 
aaa S 23 29 35 43 5) [ol TOL. [SF 13" ba 28h) ae 
m|4 35 43 BY AY OV 14 43 el ST Eb 46 2 132) 
oo. 5 O035)217 15) $51’ 81 11253 |F2 20) +8. 05.434°25 12 36 
aa eran | 
: 0 h 15"! CA hed vay hes BEY 40’ 48’’ Rye 1 08/” B O12 
of, @, | iL 20 25 32 38 46 BB G4 05\(0l 182) Tees 
2/3 | 26 33 39 47 56 (ot’.07 jo °2Z1- (2a. 38-| -2).00 
ei ie We 47 BG (ad’ GT 10k 20: |G3 36. 1b 59) 112-82. | 225 
5 | 1-07) 1’ 20 | 1’ 38 | 2 00 |.2 34 | 3 29 |.5 14 {10 16 | 
=) Oh 18) 24) . 30” = a6). 44”) 527] 17 021 1" 14") 1" 29” 
oud Be eis 0 822 28 35 42 50 | 1.00 | 112 | 1 26 | 1 45 
a Sie 11k 29 36 AB i. 52 171 O02 [1 14161 29 | 1°49 | 2) 16 
~|4 | 43 51-41’ O14) 80.43) (127-151 49.102 14 2 54/) 4°05 
OS | Vibe} WV 26 1 54 |'2 10 }-2 49 1038-55 |.6 15 |14 58 | 
Oh {| el Py ld sare 40’ 48” 57) 1 08" 1" 217 EF Bg 
D> & ue pane v 32 39 46 5? 1-4’ 06. (ed 19 41. 35.41 BOF 
S/3 33 40 48 57, | 1’ 08 |e1 21 1 88 |.2°02 | 2°36 
ee | AT 55 1,1’ 06 jo2’ 19 |a1 26: 1 58 Ke BO. [53 21 | 4 59 
5 | 115 | 31 | 151 | 2 20 | 8 05 | 4 25 | 7 34 [25 18 | 
= BOR hog 5247 oe: ante 36” 44’ 62/7140 02/1144 20 at 
a1 2 28 35 39 [Li 0-- led’ 00) g3 42-54 26° 185 1 de 
B43.) *.36 43 bo" fed'-O2, (51°43 17 P9401 - 89 (82. 27 1S 
oe; |. 50) 1700). 11 |] 26 | 1 44 1:2 10 12.49 | 3 55 | 6 16 
PS | 1d) 1.36.).1 58 | 2 30 |8 22 195 00 9 24 
) 0 h : ot 33”/ 40” HQ iy hg) ig 08’’ a A ie 39’ Q’ 2% 
oa). be %82 39 46 Boil’ 06 (04 10-85 16d. BT. ib Bee 
Se Se) 40 47 5G: (AZ OT 153 BP lel BB he DOT 2 .B4 1 Beg 
4 | 54 |1'04 | 1.16] 1 33 | 1 54 | 2 24 | 3 11 | 4 88 | 8 15 
PSay) 1°23) 1 41 | 2-05.02 41 18 40 |: 5 40 [12 02 / 
> 0 h | 30”” 36” 44’"| 5Q’’ 1 02’7| 1 14’"| ig 297") i 49”" ph 18’ 
| 2 | 35 | 42 BO hal” 00 (ed 42) Wade BED 25 182. O14 Or St 
Be Ss a5cey G1. led’ Olid 38 [19 2S, 1) 47 Pe 15 188 B6 |) SOR 
ay | 58 09 | 1 23 | 1 40 2 05 | 2 40 | 3 39 | 5 387 |11 18 
V5 | eT 1 46 12. 12 | 2 62 401 6 30 16 19 | 
| +20° | +15°| +10°| +5° | 0° | —65° | —10°} —15°| —20° 
| : ‘ } } 





OI eee ed 
a 


40 TABLE XIX.—MEAN REFRACTIONS IN DECLINATION 

















































































































































































































o 

$2 x DECLINATIONS. 

2| sa 

8 o< ° ° ° ° {e} OT ° ° ° 
H | +20° | +15° | +10 +65 it) =6° |»—10° | —15° | —20 

Oh 33” 40’ 48”’ 57/7) 17-08’) VY" 21") 1 39/7] <2” 02"7| 2% 36% 
eue 38 46 55 | 1706 | 1 18 | 1 85) 1 57 | 2 28 | 8 19 
o| 3 47 56 | 1’ 06°) 1:19 | 1 36 | 2 29 | 2 31 | 3 23 | 5 02 

4 1-02) 17-14 411-29 | 2) 48. 902 16 162°58,)°4 18°16" 597 19S 4% 

5 I 30 |-1 51 | 2 19 | 3 04 | 4 22 | 7 28 (24 10 
> Oh 36” 44” Ba 1 02") TT! 14/7) S07 ok” 40") Va! 918 eae 
a | 2 43 50 69 th 11 (tT 26-1 242%) 42598.) 9249 4 ooh 
a ik 50 | 1700 | 1’ 11 | 1 26 | 1 45 |-2 11 | 2 51 | 2 58 | 6°22 
a | 4 17.05 1 OE 18+ /RUN35 “| 82-10) | $8028 Sho: | Paeigs. ee ag 
| 5 1 34 | 1 56 | 2 27 | 3 16 | 4 47 |, 8 52 
=) | | sli ——— | — 

0 h AZ 48”’ ya |’ 08/7) 1 Palys 14 39/7 oF 02’’ oF 36” 3 33” 
oor 46 55 | 1.05 |.1 18 | 1 34 |-1 56 |-2 380 | 3°15) 4.47 
1» | 3 55, tu)? O06 (2Ee19 | 1535 121-58, 62°30) 3-21. |PaebB y) Sere 

4 PO: [BD 28. PL 42 188-006 227 43) | 28244 Gh AS toes) 

5 I 37 | 2 01 | 2 34 | 3 28 |.5 15-10 18 
s | Oh 44” Bo OL 02 OL Wa Sd? 2077) T10 a0 A 218 Pa o00 Ties 
i 2 50 59 wei Tle lieds 25: ed Pastis WOCIS2 Ayal oun Lin Cae 
svt 3 58 |/1’ 10 |:1 24 | 1 42 | 2.0% |'2 43 1.38 45 1:5. 50 [122-47 
w| 4 U7 21 22 25. ol 438 402-510 | 2. 50059355. | 6144145 49 
| 5 1 41 | 2 06 | 2 42 | 3 42 | 5 46 |12 26 

Oh 48” BT T 08h) 21770" 39" 162" 02" «2! 36“ IPS ea) Oeeoe 
ous 54 )o1’ 04) 701 °17 jel 83 |-1 54 | 2224 [98 12 |=4988 "| Sars 
©/|3 1703 j°2 15 |41 80 122 $1 [52 20 1:38.04 |°4 24 17 Bl e4eae 

4 118) 00034) 92556 2528 18818 124060 1845s 

5 D45) }O2 281) 22750" 183-267 (C6521 ° 115832 
S Oh 527 e027 1114S P29" 14? 50 lee 18! | 93! 000 | Fa ieee 
a | 2 1 09} 1 23 | 1 41 | 2 06 | 2 48 | 3 44 | 5 50 j12 44 
ey 8 LOT jedy 23422588" 82°01 08.35. ESeSOr ve abe tone4 
n| 4 1 23 | 1 40 | 2 05 | 2 40 | 3 40 | 5 87 {11 50 
215 148 |2 17 |-2 59 | 4 14 | 7 08 

Oh B78 4? 0859 21"1S139""1 6 2" 0277) 2! 86" 8538712 2310 Oke 
Baie? 1’ 03° SLC16. (S231) be 188-21 US SA0T) Me eeR oe ess 
gq 3. ed 2 Ti27 \3°46 |b2 12 92-52 154° 02 [66:38 | 

4 1 27 | 1.47 | 2 13 | 2 54 | 4 05 |'6 40 | 

5 1 52.) 2 22 | 3 08 | 4 30 | 7 52 
> 0 D2 4 a 0") Pb 507 e218" Pa 00 easel ie | eto 
4 | 2 1 08 | 1 22 | 1 40 | 2 03 ; 2 39 | 8 387 | 5 32 {11 28 
> | 3 1 17 |°1:34 | 1 55 |'2 26 | 3 14 | 4 44 | 8 34 
re | 4 1 32. |i1 538 |'2 23 |'38 14 | 4 85 |°8 05 
© 6 1 56 | 2 28 | 3 17 | 4 40 | 8 51 

Qh 1G 08" 092 217 |b. 1% 39" Set08” ieS6" FB88 ey eee" Ty OLY 
oun & Pets Gd 29) 11 BOIS 2: 18: Br O0. SST) | et 138' | 
© 3 I 23 | 1 48 | 2 05 | 2 41 | 3 41 | 5 59 |12 15 

4 1 37 82 00 (92 34 13 28 1/5 20 110 12 

5 2 02 }°2 83 |°3 27 | 5 11 |10 05 

+20° | +15° | +10°| +5° 0° —5° | —10° |) —15° | —20° 
































TABLE XX.—LENGTHS OF CIRCULAR ARCS; RADIUS=1 41 












































Sec. | Length. || Min.| Length. ||Deg.| Length. || Deg.| Length. 
1 .0000048 1 . 0002909 1 .0174533 61 1.0646508 
2 . 0000097 2 .0005818 2 .0349066 Bs 1.0821041 
3 .0000145 3 .0008727 3 .0523599 63 1.0995574 
4 .0000194 4 .0011636 4 .0698132 64 1.1170107 
5 . 0000242 5 .0014544 5 .0872665 65 1.1344640 
6 0000291 6 . 0017453 6 .1047198 66 1.1519173 
iG 0000339 vf .0020362 a 1221780 67 1.1693706 
8 . 0000388 8 0023271 8 . 1896263 68 1.1868239 
9 . 0000436 9 0026180 9 . 1570796 69 1.2042772 

10 .0000485 10 .0029089 10 . 1745829 7 1.2217305 
11 .0000533 11 .0031998 11 . 1919862 71 1.2391888 
12 0000582 12 .0034907 12 2094395 72 1.2566371 
13 .0000630 13 .0037815 13 2268928 73 1.2740904 
14 .0000679 14 .0040724 14 . 2443461 74 1. 2915436 
15 .0000727 15 0043633 15 . 2617994 5 1.3089969 
16 .Q000776 16 .0046542 16 2192527 76 1.8264502 
17 . 0000824 17 .0049451 17 2967060 TE 1.8439035 
18 .0000873 18. . 0052360 18 .8141593 78 1.3613568 
19 .0000921 19 .0055269 19 .3316126 & 1.3788101 
20 . 0000970 20 .0058178 20 .8490659 80 1.8962634 
21 .0001018 21 .0061087 21 .8665191 81 1.4137167 
22 .0001067 22 .0063995 22 8839724 82 1.4811700 
23 .0001115 23 .0066904 23 .4014257 83 1.4486233 
24 .0001164 © 24 .0069813 24 .4188790 84 1.4660766 
25 .0001212 25 .0072722 25 .4363323 85 1.4885299 
26 .0001261 26 .0075631 26 .4537856 86 1.5009832 
27 . 0001809 27 .0078540 oh 47123889 87 1.5184864 
28 .0001357 28 .0081449 28 . 4886922 88 1.5358897 
29 .0001406 29 |- .0084358 29 .5061455 89 1.5533430° 
30 .0001454 30 .0087266 30 5235988 90 1.5707963 
31 .0001503 31 .0090175 al .5410521 91 1.5882496 
32 .0001551 32 .0093084 82 .5585054 92 1.6057029 
33 0001600 33 . 0095993 33 .5759587 93 1.6231562 
34 .0001648 34 .0098902 34 .5934119 94 1.6406095 
35 .0001697 35 .0101511 35 . 6108652 95 1.6580628 
36 .0001745 36 *0104720 36 6283185 96 1.6755161 
fi -0001794 37 .0107629 37 .6457718 97 1.6929694 
38 .0001842 38 .0110588 38 .6632251 98 1.7104227 
39 .0001891 39 .0113446 39 . 6806784 99 1.7278760 
40 .0001939 40 . .0116355 40 .6981317 100 1.7458293 
41 .0001988 41 .0119264. 41 . 7155850 101 1.7627825 
42 . 0002036 42 01221738 42 .'7330383 102 1.7802358 
43 . 0002085 43 .0125082 43 . 7504916 103 1.7976891 
44 .0002133 44 .0127991 44 (679449 104 1.8151424 
45 .0002182 45 .0130900 45 (858982 105 1.8825957 
46 .0002230 46 .0138809 46 .8028515 106 18500490 
47 0002279 47 .0136717 47 .8203047 107 1.8675023 
48 . 0002327 48 .0139626 48 .8377580 108 1.8849556 
49 .0002376 49 .0142535 49 .8552113 109 1. 9024089 
50 | .0002424 50 .0145444 50 .8726646 110 1.9198622 
51 .0002473 51 0148353 51 .8901179 111 1.9373155 
52 .0002521 52 .0151262 52 . 9075712 112 1.9547688 
53 .0002570 53 .0154171 53 . 9250245 113 1. 9722221 
54 .0002618 54 .0157080 54 9424778 114 1.9896753 
55 .0002666 |) 55 .0159989 55 .9599811 115 2.0071286 - 
56 .0002715 56 .0162897 56 . 9773844 116 2.0245819 
57 . 0002763 YG .0165806 57 . 99483877 117 2.0420352 
58 . 0002812 58 .0168715 58 1.0122910 118 2. 0594885 
59 . 0002860 59 .0171624 59 1.0297443 119 2.0769418 
60 .0002909 60 .0174533 60 1.0471976 120 2.0943951 











42 TABLE XXI.—MINUTES IN DECIMALS OF A DEGREE 








4 0” 
0 .00000 
1 .01667 
2 .03333 
3 .05000 
4 .06567 
5 .08333, 
6 10000 
7 11667 
8 13333 
9 . 15000 
10 . 16667 
11 . 18333 
12 . 20000 
13 .21667 
14 . 23333 
15 .25000 
16 26667 
17 28338 
18 .80000 
19 .31667 
20 .338333 
21 . 385000 
22 .36667 
23 .38333 
24 .40000 
25 .41667 
26 .43333 
ale 45000 
28 . 46667 
29 .48333 
30 5 
31 .51667 
32 .53333 
33 .55000 
34. . 56667 
35 .58333 
36 .60000 
37 . 61667 
38 63333 
39 65000 
40 . 66667 
41 68333 
42 ft 
43 71667 
44 73333 
45 75000 
46 . 76667 
47 .78333 
48 | .80000 
49 .81667 
50 83333 
51 .85000 
52 .86667 
53 .88333 
54 . 90000 
55 . 91667 
56 . 93333 
57 . 95000 
58 . 96667 
59 . 98333 





10” 


00278 
.01944 
03611 
05275 
06944 
08611 
10278 
.11944 
13611 
15278 
. 16944 


.18611 
20278 
21944 
23611 
25278 
20944 
28611 
80278 
.381944 
33611 
385278 
36944 
38611 
.40278 
41944 
.43611 
.45278 
46944 
48611 
9027 


51944 
53611 
55278 
96944 
.58611 
60278 
61944 
63611 
.65278 
.66944 
.68611 
10278 
71944 
. 73611 
(5278 
76944 
(8611 
80278 








15" 





52222 
53889 
.59956 
57222 
.58889 


| .60556 


62222 
. 63889 
65556 
67222 
. 68889 
70556 
- (2222 
. 73889 
15556 
(222 
. 78889 
80556 





20" 








30” | 40” 





00833 | .01111 
02500 | 02778 
O4167 | 04444 
05833 | .06111 
7500 | 0777) 
09167 | .09444 
10833 | .11111 
12500 | .12778 
14167 | 14444 
15833 | 16111 
17500 | .17778 
19167 , 19444 
20833 | .21111 
22500 | 22778 
24167 | 124444 
25833 | 26111 
27500 | .27778 
29167 | .29444 
30833 | 31111 
32500 | 3277 
34167 | 134444 
35833 | .36111 
37500 | .37778 
39167 | 39444 
40833 | 41111 
42500 | 42778 
44167 | 44444 
45833 | 46111 
47500 | 4777 
49167 | .49444 
50833 | 51111 
52500 | .52778 
54167 | 54444 
55833 | .56111 
57500 | 5777 
59167 | 159444 
60833 | .61111 
62500 | 6277 
64167 | 64444 
65833 | .66111 
67500 | .67778 
69167 | .69444 
70833 | .71111 
72500 | . 72778 
74167 | .74444 
75833 | . 76111 
77500 | 777 
79167 | 79444 
80833 | .81111 
82500 | .82778 
84167 | .84444 
85833 | 86111 
87500 | .8777 
89167 | .89444 
90833 | .91111 
92500 | .9277' 
94167 | 94444 
95833 | .96111 
97500 | .97778 
99167 | .99444 
30" | 40" 











45” 


.01389 
03055 
04722 
.06389 
.08056 
09722 
11389 
13056 
14722 
. 16389 
18056 
19722 
.21389 
23056 
24722 
26389 
28056 
29722 
.31389 
33056 
84722 
.36389 
88056 
389722 
.41389 
.48056 


01250 
02917 
04583 
06250 
07917 
09583 
11250 
12917 
14583 
16250 
17917 
19583 
21250 
22917 
24583 
26250 
27917 
29583 
31250 
32917 
34583 
36250 





45" 


50" 


722 


.463889 
.48056 
.49722 

.51389 


83056 

84722 
86389 
88056 
89722 
91389 
93056 
94722 
96389 
98056 
99722 


50" 


~ as 
Scarcamwwre | = 











TABLE XXII.—TURNOUTS AND SWITCHES FROM A 


STRAIGHT TRACK 


43 


GaAuGE, 4 FEET 814 INCHES = 4.708, THRow, 5 INCHES = 0.417. 


























. | Angle | Dist. ' Switch | Radius | Log’thm.| Degree 
n. By BF. ay. PAs T. | log. r. | of Curve. 
Sees -—|—_—_—__-—| | pa akin 
4 14° 15/ 00"| 37.664] 387.3738; 11.209; 150.656 2.177986 | 38° 45/ 57” 
416 |12 40 49 | 42.3872] 42.113} 12.610 190.674 | 2.280292 | 80 24 09 
5 11 25 16] 47.080) 46.846; 14.012 235.400 | 2.371806 | 24 31 36 
51g | 10 23 20} 51.788] 51.575] 15.413 284.834 | 2.454592 | 20 13 13 
6 9 31 30 | 56.496| 56.301} 16.814 838.976 | 2.580169 | 16 57 52 
644 | 8 47 51 61.204 | 61.024} 18.215 397 826 2.599693 | 14 26 25 
7 8 10 16 65.912] 65,744; 19.616 461.884 | 2.664063 | 12 26 34 
“46 | ¥ 37 41) 70.620] 70.464] 21.017 529.650 | 2.723989 | 10 50 02 
8 |G 09 10 75.3828 | 75.181] 22.418 602.624 2.780046 | 9 31 07 
8144 | 6 43 59 | 80.036] 79.898} 23.820 680.306 | 2.882704} 8 25 47 
9 6 21 35 84.744) 84.618] 25.221 762.696 2.882352 | 7 31 04 
916 | 6 01 32 | 89.452] 89.3828] 26.622 849.794 | 2.929814 | 6 44 46 
10 5 43 29} 94.160] 94.043] 28.023 941.600 |. 2.978866 | 6 05 16 
10% | 5 27 09 98.868} 98.756 | 29.424) 1088.114 3.016245 5 dl 17 
11 5 12 18 | 103.576 | 103.469 | 30.825] 1139.386 3.056652 5 01 50 
11% | 4 58 45 | 108.284] 108.182 | 32.227] 1245.266 3.095262 | 4 36 08 
12 4 46 19 {| 112.992 | 112.894 | 383.628] 1355.904 3.132229 | 4 13 36 
GavuGE, 3 Feet. Turow, 4 IncHEs = 0.333. 
No: Angle Dist. -| Chord | Switch; Radius | Log’thm. | Degree 
n. F. BF, af. AD. ie, log. r. of Curve. 
14° 15/ 00" 24 23.815 8 96.0 1.982271 62° 467 34" 
41g | 12 40 49 Ke 26.835 9 121.5 2.084576 | 48 36 04 
5 it<25016 30 29.851 10 150.0 2.176091 | 38 56 33 
514 | 10 23 20 33 32.865 11 181.5. | 2.258877 | 31 58 55 
6 9 31 39 36 35.876 12 216.0 2.384454 26 46 07 
644 8 47 51 39 38.885 13 253.5 2.403978 22 45 04 
7 8 10 16 42 41.893 14 294.0 2.468347 19 35 O1 
vers v7 37 41 45 44.900 15 337.5 2.528274 | 17 02.21 
8 7 09 10 48 47.906 16 384.0 2.584331 | 14 57 48 
844 6 43 59 51 50.912 17 433.5 2.636989 | 13 14 47 
9 Ge2te 3h 54. 53.917 18 486.0 2.686636 | 11 48 37 
946 6 O1 82 57 56.921 19 541.5 2.733598 10 35 46 
10 5 43 29 60 59.925 20 600.0 2.778151 9 33 38 
10% 5 27 09 63 62.929 21 661.5 2.820530 8 40 12 
11 5.12718 66 65.932 22 726.0 2.860987 7-58 O4 
1144 |) 4 58 45 69 68 . 935 23 793.5 2.899547 7 13 82 
12 4 46 19 72 71.938 24 864.0 2.936514 6 38 06 
ANGLE AND DISTANCE OF MIDDLE FRoG, ff" 
Gauge | Gauge Gauge | Gauge 
No.| No. | Angle | 4, 8%. 3, No.| No. | Angle | 4, 84. 3. 

n, n" 3 eee Dist. Dist. n. n", Mee Dist. Dist. 
al!) ak", GH" S\eak”. 
4 | 2.817) 20° 07’ 36"| 26.736 | 17.037 || 8 | 5.651 | 10° 06’ 44”| 58.317 | 33.974 
4¥4| 3.172, 17 54 52s 30.054 | 19.151 || 814) 6.005| 9 381 08] 56.643 | 36.094 
5 | 3.527116 08 19| 33.374 | 21.266 9 | 6.359) 8 59 80) 59.969 | 38.213 
544) 3.881); 14 40 58) 36.695 | 23.388 914| 6.713 | 8 81 10} 63.296 | 40.333 
6 | 4.235) 13 27 57 | 40.018 | 25.500 || 10 7.067 | 8 05 40} 66.623 | 42.453 
64g) 4.589) 12 26 07 | 43.342 | 27.618 | 1014! 7.420) 7 42 35] 69.950 | 44.573 
(i 4.943) 11 33 041! 46.666 | 29.736 || 11 7.774 | 7 21 36) 73.277) 46.693 
Te) 5.297) 10 47 02} 49.991 | 31.855 || 1114) 8.128 | 7 02 26) 76.605} 48.813 
8 | 5.651| 10 06 44| 53.317 | 33.974 2 | 8.482| 6 44 51] 79.932 | 50.984 








Chord 





































































































GINEERING 


EN 
TURNOUTS 


SPLIT SWITCH 


TABLE XXIIA.—AMERICAN RAILWAY 
ASSOCIATION. 


44 






























































LL OST | 99° OST | 60°SLT | sr 60 T LL’ ee6y.| G ¢ 0 O2'S€ L6°0 | L7°0 9 7 Ci bel. Bete SLs S60 FG 
9L°EIL | 89°SIl | Ge'9ST | ce Gh T | 86°e9ce| G eS 0 ONee 1604 “87 04-9 C6e- 1 Fo OT Be Ig IG ¢@ 02 
19° FOL | FS POT | 8E°9FT | 2 GI SB | 99'48Ge, G eo DO 0 €8°'|-86°0 | 6F°0 | (9 92} 8 AL) OT 8 9¢ OI € 81 
GO°S6 =| 96°F%6 G6°SeI | Fe 1G @ g6°s00g| § 2g 0 0 €€ | 0O'T | 09°0 0 Fe OF OL; 0.8 LY VE € 9T 
6°68 | €8°68 | 0S 0&1 | 90 LT & GP PPL G 6G 0 0 .€& 66°0 | 19°0 9 66 Oterlcs cee 90 67 € GT 
96°89=./ 88°89 | G2°46 | 86 20 9. | Fe 98IT; 8 slat 0 22°} TO°L |} 89°70 TS ca meer I oe) 61 9F 7 rat 
0¢° 79 | 90°79 90°36 8y GO 9 12'0v6 | 8 8t fT 0 G6 GO°T | ¥S°0 9 LT OP ite ee oO 8I el & II 
LISS =| 10°99 TG LL 8I Gt 24 G2°06L |. IT fr I ORgt GO°T | 09°0 9=91 SS COT eOne 9 6c &F OT 
8G°eS | 0F' co | 06° FL Cott 8 L6°669 | IL ¥F T 9 91 | SOL | £9°0 OBOE: OmeOle On 29 ee 10 9 | #6 
66°6h | FL 6h | Foch Le 81 6 Le 919 Il *F TI 9 OL | Leb) 9) 10 0 9T CeeOl SROGe o cé Ie 9 6 
ov 9F | 36c°9F | LY LO OF PF IL | [L887 Tir 9° OL. +) 60.- T1690 ey tg) 6 8 Cie OI 60 L 8 
ve 1h | G01? v6°19 «| 6h LY GT | 88° V9E Il v7 T 9.291 of <SE--L +}. 669"0 9 GI teas CRT. 9I OI 8 h 
SE °eé> | Tl ee II 87 8¢ 10 12 | G6°€4¢ | 61 9E @ 0 11 | 9I'T |] 99°0 Omeub 0 2 Or je ee Te°6 9 
GG 8c | 61°82 LL’ GP bop0r olf) con S8L |. 61, 9S" 6 0 Il 86°I | 14 °0 0 OL Gro Lees 91 Se If | ¢ 
62°22 18822 |,GO°LE 1,99 86 o&$ | 92 SIT | .6T 98 08 | «OTT | 28°T | 640) 9 8 [uF & |e & |.00 ST Fl | F 
ee a a | i a a 
Q Q ae i) a R Sa wp wR 4 = “ my = 
S 5 e. r en f AL Re 2 Dl a asfar . l 
@ Bed ret ay be y 2 g g p Sree ree a iS 
a a Oro. Eo ee 3, sal a A H aS 58 3 3 
Bee. 28 Push oe © © > Bog o OF 08 0 
Q @ Bin o ® 7 a ea a - = 5 a 
S = ofa ric 09 oe ° - a 
q 2 mo S, ° > o ae | ae 4 5 c 
< a a + ° mr @) ° iS a a oR 
o 08 mao so Q 5 m oO ® : ene orn 5 
as = oor sy . & 4 2 83 29 : g 
Re eae © = ° Es ae 3 n 
eas a 2. Qu ® S ct. ‘Se 
Lr = | or. 3 S. 
"19 = Y= aourqsiq 
; bigs a “wiO= pies q 3 A 
ulod jo sseuyoryy, = SyUlog Borg [TV Jo ssouyoryy, 
a ee | soyoqyTag TLV Jog ‘SDOUd AO SHILUACOUd 
| ‘SHHOLIMS 
'\fO SHILYAdOUd 














45 


TABLE XXIIB.—AMERICAN RAILWAY ENGINEERING 


SPLIT SWITCH TURNOUTS 


ASSOCIATION, 












































00°€ (6S CET | L6°T [Te OOT | Le'T \28°L9 e8x7 SEX €+ 68° SE Ze LAT | B2°9LT | 00'0 | EFS | Te OT I | 91 '988h| Fe 
€6°¢ |69 8IT | 88° |12'06 | 80'T |P8 "19 |\Le¢x €+ 00'S |€E+ L2X 6+ 26°92) Sh LGT | 6G°9ST | 00°0 | F7'0 | CE SF I | 92° LGCE) 02 
98°¢ JOT OIT | c8°1 97 F8 | FO'T |€4°8S 9X P 9¢X E+ £696 TG'9FT | 9L°SPT | 80°T | 00°0 | TE FI @ | TE 9FGe| 8T 
} 5. St een SERS Os a a ce > | -__- 
L8°@ |SE°SOT | c8'T 94°18 | POT /9T° 8S |€€X + 00 '0E eX 6+ 06°62 LG'LET | 06°9ET | 00°0 | 9G°T | 6S eG @ | be E661; YT 
v8°S |S OOT | 8L4°T |86°LL | 10'T [67 SS 0€X € O€X 6+ 68°62 GI IST | 99°O8T | 00°0 | 60'0 | OT LT €& | O8'EPLT| ST 
16'S |86° LL | O6'T |G9°09 | GI'T \66°€7 eX & VOX 6+ 88'ES 08 OOT | OF OOL | 00°0 | €€°S | 6G SI G | €L°860T) GT. 
L8°G |6T' 2h | 78°T |Lb'9S | 80'T IFL‘0F EX @ €€+ G8 ° SE) 1816 G8°6 | 00°0 | 66°2| Le eI 9 | G9°ce6 | IT 
G8°o |I8 LG | F8°T |G0° FF | 90°T \82'0€ 86+ LI LE 86+ 00°12) §6° LL Tg'LL | 00°0 |00°0 | 8T ST 24 | $6°06L | OL 
€8°¢ |LE'9G | c8'T |G eh | 90'T |TE'0€ LéE+ 00 92 Lé+ 08° G2) TL’ Sh 6°GL |00°0 | 9L°0 | Sh FI 8 | Sh°S69 | 76 
GL°e@ |61'€G | 9L°T |86'°0F | 2O'T [GL °82 €€+ 69 9T €€+ IF 9T| 82 °eh 16°TL | 26°0 |00°0 | ch 82 6 | 81'909 | 6 
16°S |S7'1G | 8L°T |16°6E | ZO'T |Le°82 0&+ 09 9T 0€+ 09 91} 86°19 G9°L9 | 00°0 | 08°0 | Le 9F IT] 87° L8F | 8 
bL°@ |IL Ly | TL'T |€6°9E | L6°0 \2L°92 Le+ Il‘ FT Lé+ 68° €T| OT’ 39 18°19 61°0 | 00'0 | 62 eG GT) 80°c9E | & 
oL'@ |ST°GE | PLT ST Le | 10'T |L0 61 €€ €L G8! 86 LP €L°L47 |99°0 |00°0 | ¥0 SF Te| 6€°G9e | 9 
69's [Le 18 | 19°T |PS° FS | G6°0 |8L LT 86 89° Lé| LY 3P 9¢°cr | 68'0 |00'0 | LG 6I &8| Pe PLT | G 
6L°6 |GL°66 | LOT \Ph se | L6°0 bh LT VG 09°€6| VE LE | LL LE |,00'0 |80°T |,76 2 o€|,69 OIT | F 
| a A ak lect 
= Re SS. ae ES ram be, a Q Q S) Oe a eo S) Aes 
Oe wg P22) 24288 | ae |. Oe ] ese lg 
$ eS al Vous ise ull. caer Sos a rk 
‘UISIICQ) 8B [IVY YouIMY jo Juiog & g, E> 8 oS b> re | eg SB be Z 
0} padiojoy ‘[Tey paaimg jo opis © ee ope et ees Be a et an 6. Sh 5 
esney uo sjyulog J0}Ua,) JoJIeN? 5 < Be Soe eS le = 0 e 
04} OF soqvUIpPIO-o9 iIvNFueW07T 3 fa 5a SEE ® g C 2 ® 
= = Pea hd Bad + o io = 
oe. ao 9, + t = © 
B B | 8 





























SaVaT TVOILOVUd 





46 


TABLE XXIII.—SQUARES, CUBES, SQUARE ROOTS 





~~ 


No. 


OMI CW eH 


ge 


Squares. 





Cubes. 


117649 


125000 
132651 
140608 
148877 
157464 


166375 


175616 
185193 
195112 
205379 


216000 
226981 
238828 





Square 
Roots. 


1.0000000 
1.4142136 


2 4494897 
2.6457513 
28284271 
3 .V000000 
3.162277 

3.3166248 
3 .4641016 
8.6055513 
3.7416574 
3.8729833 
4. 0000000 
4.1231056 
4 2426407 
4.3588989 
44721360 
4. 5825757 
46904158 
4.7958315 
4. 8989795 
5.0000000 
5 0990195 
5. 1961524 
52915026 
5.3851648 


5.472256 
5 5077644 
56568542 
5. 7445626 
5.8309519 
5.9160798 
6.000000 
60827625 
6. 1644140 
6.2449980 
63245553 
6. 4031242 
64807407 
6.5574385 
6.6332496 
6.7082039 
6.7823300 
68556546 
6. 9282082 
70000000 


70710678 
71414284 
72111026 
72801099 
7 .B484692 
74161985 
74833148 
75498344 
76157731 
* 76811457 
"1459667 
78102497 
78740079 











Cube Roots. 


1.0000000 
1.2599210 
1. 4422496 
1.5874011 
1.7999759 
1.8171206 
1.9129312 
20000000 
2.0800837 


~ 2.1544847 
2 2239801 
22894286 
2.35183847 
2.4101422 
2.4662121 
2.5198421 


2 


2.5712816 - 


2.6207414 
2.6684016 


2.7144177 
2.7589243 
2.8020393 
2.8488670 
2.8844991 
2.924017 
2.9624960 
3.0000000 
3 .0865889 
8 .0723168 


3. 1072825 
3.1418806 
3.1748021 
8.20753843 
3.2696118 
3.2710663 
8 .8019272 
3.3322218 
3.3619754 
3.3912114 


3.4199519 
3.4482172 
3.4760266 
3.5033981 
3.53803483 
3.5568933 
3.5880479 
3. 6088261 
3 .6842411 
3.6593057 


3.6840314 
8. 7084298 
8. 7825111 
8.7562858 
3.7797631 
8. 8029525 
3 .8258624 
3.8485011 
3. 8708766 
8 .8929965 


8.9148676 
8. 9364972 
3.9578915 





Reciprocals. 


1.000000000 


.500000000 
. 333333333 
- 200000000 
- 200000000 
- 166666667 
142857143 
- 125000000 
-111111111 


- 100000000 
.090909091 
083333333 
-076923077 
071428571 
. 066666667 
. 062500000 
058823529 
055555556 
052631579 
050000000 
047619048 
045454545 
043478261 
041666667 
.040000000 
-038461538 
037037037 
085714286 
- 034482759 


. 033333333 
0382258065 


029411765 
028571429 
027777778 
-027027027 
-026315789 
025641026 


-025000000 
024390244 
. 023809524 
028255814 
022727273 
022222222 
.021739130 
021276600 
-020833333 
020408163 


020000000 
019607843 
-019230769 
.018867925 
.018518519 
018181818 
017857143 
017548860 
017241379 
016949153 


016666667 
016393443 
016129082 


CUBE ROOTS, AND RECIPROCALS 


47 





Squares. 


3969 
4096 
4225 
4356 
4489 
4624 
4761 


14400 
14641 
14884 
15129 
15376 








Cubes. 


250047 
262144 
274625 
287496 
300763 
314432 
828509 


343000 
857911 
873248 
389017 
405224 
421875 
438976 
456533 
474552 
493039 
512000 
531441 
551368 
571787 
592704 
614125 
636056 
658503 
681472 
704969 


729000 
53571 
778688 
804357 
830584 
857375 
884736 
912673 
941192 
970299 

1000000 

1030301 
1061208 
1092727 
1124864 
1157625 
1191016 
1225043 
1259712 
1295029 


1331000 
1367631 
1404928 
1442897 
1481544 
1520875 
1560896 
1601613 
1643032 
1685159 


1728000 
1771561 
1815848 
1860867 
1906624 














Square 
Roots. 





7. 9372589 
8 .0000000 
8.062257 
8.1240384 
8.1853528 
8. 2462113 
8.3066239 
8.3666003 
8. 4261498 
8.4852814 
8.5440037 
8.6023253 
8.6602540 
8.717979 
8.749644 
8.8317609 
8.8881944 


8.9442719 
90000000 
9 .0553851 
9.1104836 
9.1651514 
9.2195445 
9. 2786185 
9.38273791 
9 .3808315 
9 4339811 


9. 4868330 
9 5393920 
9 .5916630 
9 .6486508 
9.6953597 
9.'7467943 
9.7979590 
9.8488578 
9.8994949 
9.9498744 


10.0000000 
10.0498756 
10.0995049 
10.1488916 
10.1980390 
10.2469508 
10.2956301 
10.3440804 
10.3923048 
10.4403065 


10.4880885 
10.53565388 
10.5830052 
10. 6801458 
10.6770783 
10. 7238053 
10.7703296 
10.8166538 
10. 8627805 
10.9087121 


10.9544512 
11.0000000 
11.0453610 
11.0905365 
11.1855287 





Cube Roots. 


3.9790571 
4.0000000 
40207256 
4.0412401 
4.0615480 
4.0816551 
4.1015661 


4.1212853 
4.1408178 
4.1601676 
4.1793390 
4.1983364 
4.2171633 
4.2358236 
4 2548210 
4 2726586 
4.2908404 
43088695 
4 8267487 
43444815 
4.3620707 
4.3795191 
4 3968296 
4.4140049 
4 .4310476 
4.4479602 
4.4647451 


4.4814047 
4.4979414 
4.5148574 
45306549 
45468359 
45629026 
4.5788570 
4.5947009 
4.6104363 
4.6260650 


4.6415888 
46570095 
46723287 
4. 6875482 
47026694 
4.7176940 
4.326235 
4.7474594 
47622032 
4.768562 


4.7914199 
4. 8058955 
48202845 
4.8345881 
4.8488076 
4.8629442 
4.8769990 
4.8909732 
4.9048681 
4.9186847 


4. 9324242 
»4. 9460874 
4.9596757 
4.9731898 
4. 9866310 





Reciprocals. 


.015873016 
.015625000 
015384615 
.015151515 
0149253873 
.014705882 
.014492754 


.014285714 
.014084507 
013888889 
.013698630 
0135138514 
013333333 
.013157895 
012987013 
012820513 
012658228 


.012500000 
.012845679 
012195122 
.612048193 
011904762 
.011764706 
.011627907 
011494253 
011363636 
011235955 


.011111111 
.010989011 
010869565 
.010752688 
.010638298 
0105263816 
010416667 
.010309278 
.010204082 
.010101010 


.010000000 
.009900990 
009803922 
.009708738 
.009615885 
009523810 
009433962 
.009345794 
.009259259 
.009174312 


009090909 
.009009009 
008928571 
.008849558 
.008771930 
.008695652 
.008620690 
008547009 
.008474576 
008403361 


.008333333 
008264463" 
.008196721 
.008130081 
* ,008064516 





48 








TABLE XXIII.—SQUARES, CUBES, SQUARE ROOTS 











No. | Squares. 
125 15625 
126 15876 
127 16129 
128 16384 
129 16641 
130 16900 
131 17161 
182 17424 
1383 17689 
134 17956 
135 18225 
136 18496 
137 18769 
138 19044 
139 193821 
140 19600 
141 19881 
142 20164 
143 20449 
144 20736 
145 21025 
146 21316 
147 21609 
148 21904 
149 22201 
150 22500 
151 22801 
152 23104 
153 23409 
154 23716 
155 24025 
156 24336 
157 24649 
158 24964 
159 25281 
160 25600 
161 25921 
162 26244 
163 26569 
164 26896 
165 27225 
166 27556 
167 27889 
168 28224 
169 28561 
170 28900 
171 29241 
172 29584 
173 29929 
174 30276 
175 30625 
76 30976 
177 313829 
178 31684 
179 32041 
180 32400 
181 32761 
182 33124 
183 33489 
184 33856 
185 34225 
186 34596 





Cubes. 


1953125 
2000376 
2048383 
2097152 
2146689 


2197000 
2248091 
2299968 
2352637 
2406104 
2460375 
2515456 
2571353 
2628072 
2685619 


2744000 
2803221 
2863288 
2924207 
2985984 
3048625 
8112136 
3176523 
8241792 
8307949 


8375000 
3442951 
8511808 
8581577 
8652264 
8723875 
3796416 
8869893 
8944312 
4019679 


4096000 
4173281 
4251528 
4330747 
4410944 
4492125 
4574296 
4657463 
47416382 
4826809 


4913000 
5000211 
5088448 
5ITT717 
5268024 
5359375 
5451776 
5545233 
5639752 
5735339 


5832000 
5929741 
6028568 
6128487 
6229504 
6331625 
6434856 


Square 
Roots. 


11. 1803399 
11 .224972% 
11. 2694277 
11.3137085 
11.3578167 


11.4017548 
11. 4455231 
11. 4891253 
11.5825626 
115758869 
11.6189500 
11.6619038 
11.'7046999 
11.7473401 
11.7898261 


11.8321596 


© 11.8748421 


11.9163753 
11. 9582607 
12.0000000 
12.0415946 
120830460 
12. 1243557 
12.1655251 
12.2065556 


12.2474487 
122882057 
12.3288280 
123693169 
12 4096736 
12. 4498996 
12.4899960 
12.5299641 
12.5698051 
12. 6095202 


12.6491106 
12.6885775 
12.7279221 
12. 7671453 
12. 8062485 
12.8452326 
12.8840987 
12. 9228480 
12.9614814 
13.0000000 


13.0384048 
13.0766968 
13.1148770 
13. 1529464 
13. 1909060 
13.2287566 
13. 2664992 
13.3041847 
13.3416641 
13.3790882 


13.4164079 
13.4536240 
13.4907376 
135277498 
13.5646600 
13.6014705 
13.6381817 








Cube Roots. 


5.0000000 
50182979 
5 0265257 
5.0396842 
5.052743 


5.0657970 
5.0787531 
5.0916434 
5.1044687 
5.1172299 
5.1299278 
5. 1425632 
51551367 
5.1676493 
5.1801015 


5.1924941 
52048279 
5.2171034 
52298215 
5 2414828 
52535879 
5 2656374 
52776321 
5. 2895725 
5.38014592 


5 3132928 
5.3250740 
53368033 
58484812 
58601084 
5.3716854 
5.3882126 
5 3946907 
5 .4061202 
5.4175015 


5 4288352 
5.4401218 
5.4513618 
54625556 
54737037 
5.4848066 
5 4958647 
55068784 
5.5178484 
55287748 


55396583 
5.5504991 
5.5612978 
5.5720546 
55827702 
55934447 
5.6040787 
5.6146724 
6 .6252263 
5. 6357408 


5 .6462162 
56566528 
5.6670511 
5.674114 
~ 5.6877340 
5. 6980192 
5..7082675 





Reciprocals. 


.008000000 
-007936508 
.007874016 
-007812500 
007751938 
.007692308 
.007633588 
.007575758 
-007518797 
.007462687 
.007407407 
007352941 
-007299270 
-007246377 
.007194245 
.007142857 
.007092199 


006896552 
.006849315 
.006802721 
-006756757 
-006711409 - 


-006666667 
.006622517 
.006578947 
.006535948 
006493506 
.006451613 
.006410256 
.006369427 
.006329114 
006289308 


-006250000 
.006211180 
006172840 
.006134969 
.006097561 
.006060606 
-006024096 
.005988024 
005952381 
-005917160 


.005882353 
005847953 
.0058138953 
005780347 
-005747126 
-005714286 
-005681818 
-005649718 
005617978 
-005586592 


005555556 


.005376344 





—_—_—_—— 
| a | 


are 








CUBE ROOTS, AND RECIPROCALS 


Squares. 


Cubes. 


6539203 
6644672 
6751269 
6859000 
6967871 
7077888 
7189057 
7301384 
7414875 
7529536 
7645373 
7762392 
7880599 
8000000 
8120601 
8242408 
8365427 
8489664 
8615125 
8741816 
8869743 
8998912 
9129329 


9261000 
9393931 
9528128 
9663597 
9800344 
9938875 
10077696 
10218313 
10360232 
10503459 


10648000 
10793861 
10941048 
11089567 
11289424 
113890625 
11543176 
11697083 
11852352 
12008989 


12167000 
12326391 
12487168 
12649337 
12812904 
12977875 
13144256 
13312053 
13481272 
13651919 


13824000 
13997521 
14172488 


** 14348907 


14526784 
14706125 
14886936 
15069223 
15252992 


Square 
Roots. 


13.6747943 
13.71138092 
13. 7477271 


13.7840488 
138202750 
13. 8564065 
13.8924440 
13.9283883 
13 9642400 
14.0000000 
14.03856688 
14.0712473 
14.1067360 


14.1421356 
14.1774469 
14.2126704 
14.2478068 
14.2828569 
14.3178211 
14.3527001 
14.3874946 
144222051 
14. 4568823 


14.4913767 
14.5258390 
14.5602198 
14.5945195 
14. 6287388 
14.6628783 
14.6969385 
14.73809199 
14.7648231 
14.7986486 


14.8323970 
14. 8660687 
14.8996644 
14.9831845 
14.9666295 
15 ..0000000 
150832964 
15 0665192 
15.0996689 
15.1827460 


15.1657509 
15.1986842 
15.23815462 
15 ..2648375 
15.2970585 
15.8297097 
153622915 
15 .8948043 
154272486 
15. 4596248 


154919334 
155241747 
15 5563492 
15.5884573 
15 .6204994 
15. 6524758 
15.6843871 
15.7162336 
15 .7480157 





Cube Roots. 


5.7184791 
5.7286543 
5. 7387936 


5.7488971 
5.7589652 
57689982 
5 7789966 
5 .'7889604 
5.7988900 
58087857 
5.8186479 
58284767 
58382725 


58480355 
5.8577660 
5.8674643 
5.8771807 
5. 8867653 
5 8963685 
59059406 
5. 9154817 
59249921 
5.9344721 


5 9439220 
5. 9583418 
5. 9627820 
5. 9720926 
5.9814240 
5. 9907264 
60000000 
6 0092450 
6.0184617 
6.0276502 
6 .0368107 
60459485 
6 .0550489 
6 0641270 
6 .0731779 
6 0822020 
6 .0911994 
6. 1001702 
6.1091147 
6.11803832 


61269257 
6.1857924 
6.1446837 
6.1534495 
6.1622401 
6.1710058 
6.1797466 
6. 1884628 
6.1971544 
6.2058218 


6 .2144650 
6. 2230843 
6 .2316797 
62402515 
6. 2487998 
6 .2573248 
6 2658266 
6.2743054 
6.2827613 


49 


Reciprocals. 


.005847594 
-0053819149 
005291005 


.005263158 
005285602 
-005208333 
-005181847 
-005154639 
.005128205 
-005102041 
005076142 
-005050505 
005025126 


.005000000 
-004975124 
-004950495 
004926108 
.004901961 
.004878049 
004854369 
.004830918 
.004807692 
.004784689 


004761905 
004739336 


716981 


004694836 
.004672897 
.004651163 
.004629630 
004608295 
004587156 
.004566210 


004545455 
004524887 
.004504505 
004484305 
.004464286 
.004444444 
004424779 
.004405286 
004885965 
004366812 


.004347826 
.004329004 
004310345 
004291845 
.0042738504 
.004255319 
004237288 
.004219409 
004201681 
004184100 


.004166667 
004149378 
.004182231 
004115226 
004098361 
004081633 
004065041 
004048583 
004032258 


SS, 


50 TABLE XXIII.—SQUARES, CUBES, SQUARE ROOTS 














No Squares. Cubes. pauate Cube Roots. | Reciprocals. 
249 62001 15438249 | 15.7797338 6.2911946 004016064 
250 62500 15625000 158113883 6.2996053 . 004000000 
251 63001 15813251 158429795 6.38079935 .003984064 
252 63504 16003008 15 .8745079 6.3163596 .003968254 
253 64009 16194277 15.9059737 6.38247035 .008952569 
254 64516 16387064 15.9373775 6 .3330256 -003937008 
255 65025 16581375 15, 9687194 6.3418257 -003921569 
256 65536 16777216 16 .0000000 63496042 .003906250 
257 66049 16974593 16.0312195- 6 .3578611 .003891051 
258 66564 17173512 16.0623784 4 6.8660968 .003875969 
259 67081 17373979 160934769 6.38743111 .003861004. 
260 67600 17576000 16.1245155 6..3825043 .003846154 
261 68121 17779581 16.1554944 6.3906765 .003831418 
262 68644 17984728 16.1864141 6.3988279 .003816794 
263 69169 18191447 16.2172747 6. 4069585 003802281 
264 69696 18399744 162480768 6.4150687 003787879 
265 G0225 18609625 16.2788206 6.4231583 003773585 
266 70756 18821096 16 38095064 6 4312276 .003759398 
267 71289 19034163 16.3401346 64392767 .003745318 
268 71824 19248832 163707055 6 4473057 .003731343 
269 72361 19465109 164012195 64553148 .003717472 
paid 72900 19683000 16 .4316767 64633041 .003703704 
271 (3441 19902511 16 .4620776 6.4712736 .003690037 
272 73984 20123648 16. 4924225 64792236 .003676471 
273 74529 20346417 16 5227116 6.4871541 003663004 
27 75076 20570824 165529454 6.4950653 .003649635 
R20 75625 20796875 16.5831240 6.502957 003636364 
276 76176 21024576 16.6182477 6.5108300 .003623188 
277 76729 21253933 16. 6433170 6.5186839 .008610108 
27 77284 21484952 16. 6733320 6.5265189 .003597122 
279 T7841 21717639 16.7082931 6.5343351 .003584229 
280 78400 21952000 16.'7332005 6.5421326 .003571429 
281 78961 22188041 16. 7630546 6.5499116 .003558719 
282 79524 22425768 16. 7928556 6.5576722 .003546099 
283 80089 22665187 168226038 6.5654144 .003583569 
284 80656 22906304 16.8522995 6.5731385 .003521127 
285 81225 23149125 16.8819430 65808443 .0035087'72 
286 81796 23393656 16.9115345 6.5885323 .003496503 
287 82369 23639903 16.9410743 6 5962023 .003484821 
288 82944 23887872 16 .9705627 6 .6038545 .003472222 
289 83521 24137569 170000000 6.6114890 003460208 
290 84100 24389000 170293864 6 .6191060 . 003448276 
291 84681 24642171 17.0587221 6.6267054 .003436426 
292 85264 24897088 17. 0880075 6 6342874 -003424658 
293 85849 25153757 17.1172428 6.6418522 .003412969 
294 86406 25412184 17. 1464282 66493998 .003401861 
295 87025 25672375 17,1755640 6..6569302 .003389821 
296 87616 25934336 17. 2046505 6.6644437 .003378378 
297 88209 26198073 17.2336879 6.6719403 .003367003 
298 88804 26463592 172626765 6 6794200 .003855705 
299 89401 26730899 172916165 6.6868831 003344482 
300 90000 27000000 173205081 66943295 .003333333 
301 90601 27270901 178493516 6.7017593 -003322259 
802 91204 27543608 17.3781472 6. 7091729 .003311258 
303 91809 27818127 174068952 6.7165700 . 003300330 
304 92416 28094464 17.4355958 6.7239508 003289474 
305 93025 28372625 17 4642492 6.7313155 .003278689 
306 93636 28652616 174928557 6.7386641 .0038267974 
307 94249 28934443 175214155 6.7459967 | - .0038257329 
308 94864 29218112 17.5499288 6.7533134 .003246753 
309 95481 29503629 175783958 6.7606143 .003236246 
310 96100 29791000 17.6068169 6. 7678995 003225806 
































CUBE ROOTS, AND RECIPROCALS 


51 











Squares, 


101761 


102400 
103041 
103684 
104329 
104976 
105625 
106276 
106929 
* 107584 
108241 
108900 
109561 
110224 
110889 
111556 
112225 
112896 
113569 
114244 
114921 


115600 
116281 
116964 


117649. 


118336 
119025 
119716 
120409 
121104 
121801 


122500 
123201 
123904 
124609 
125316 
126025 
126736 
127449 
128164 
128881 
129600 
130321 
131044 
131769 
132496 
133225 
133956 
134689 
135424 
136161 
136900 
137641 
138384 


2 








Cubes. | 


30080231 
30371328 
30664297 
30959144 
31255875 
31554496 
31855013 
32157432 
32461759 
32768000 
33076161 
33386248 
33698267 
34012224 
34328125 
34645976 
34965783 
35287552 
35611289 
35937000 
36264691 
36594368 
36926037 
37259704 
87595875 
37933056 
38272753 
38614472 
88958219 


39304000 
39651821 
40001688 
40353607 
40707584 
41063625 
41421736 
41781923 
42144192 
42508549 


42875000 
43243551 
43614208 
43986977 
44361864 
44738875 
45118016 
45499293 
45882712 
46268279 


46656000 
47045881 
47437928 
47832147 
48228544 
48627125 
49027896 
49430863 
49836032 
50243409 


50658000 
51064811 
51478848 








Square 
Roots. 


17.685192 

176685217 
17.6918060 
17.7200451 
17.7482393 
17.7763888 
178044938 
17 ..8825545 
17.8605711 


178885438 
17.9164729 
179443584 
179722008 
18 .0000000 
18 .0277564 
18.0554701 
18.0831413 
18.1107708 
18.1383571 


18. 1659021 
18. 1934054 
18. 2208672 
18. 2482876 
18.2756669 
18.3030052 
183303028 
1838575598 
183847763 
18.4119526 
184390889 
18. 4661853 
18 4932420 
185202592 
185472870 
18.5741756 
18.6010752 
18. 6279360 
186547581 
18.6815417 


18. 7082869 
18 .7349940 
18.7616630 
187882942 
18.8148877 
18 .8414437 
188679623 
18. 8944436 
18. 9208879 
18. 9472953 


18. 9736660 
19.0000000 
19.0262976 
19.0525589 


_ 19.0787840 


19. 10497382 
19. 1811265 
19. 1572441 
19.1838261 
19.2093727 


19. 2353841 
192613603 
19.2873015 





Cube Roots 


6.7751690 
6 .7824229 
6.7896613 
6.'7968844 
6.8040921 
6.8112847 
6.8184620 
6. 8256242 
6. 8327714 


6 .8399037 
6.8470213 
68541240 
6.8612120 
68682855 
6.8753443 
6 .8823888 
6.8894188 
68964345 
6. 9084359 


6. 9104232 
6.9173964 
6. 9243556 
6. 9313008 
6.9882321 
6 .9451496 
6.9520533 
69589434 
6. 9658198 
6.9726826 
6.97953821 
6.9863681 
6. 9931906 
7 .0000000 
70067962 
7 01385791 
7 .0203490 
7.0271058 
7 0338497 
70405806 


70472987 
7.0540041 
7 .0606967 
70673767 
70740440 
7 0806988 
70873411 
7 0939709 
71005885 
7.1071987 


7.1187866 
71208674 
7.1269360 
@ 1834925 
71400370 
7. 1465695 
7. 1530901 
71595988 
7 .1660957 
71725809 


71790544 
7.1855162 
71919663 





. | Reciprocals. 





008215434 
008205128 
003194888 
.003184713 
.003174603 
003164557 
.003154574 
.003144654 
.008134796 
003125000 
003115265 
.003105590 
003095975 
003086420 
003076923 
003067485 
003058104 
003048780 
.008089514 


003030303 
003021148 
008012048 
903003003 
002994012 
002985075 
.002976190 
.002967359 
002958580 
002949853 


.002941176 
.0029382551 
.002923977 
002915452 
002906977 
.002898551 
002890173 
.002881844 
002873563 
002865330 
002857143 
002849003 
002840909 
.002832861 
.002824859 
002816901 
.002808989 
002801120 
002798296 
002785515 
002777778 
002770088 
002762431 
002754821 
002747253 
.0027389726 
002782240 
-.002724796 
.002717391 
002710027 
002702708 
.002695418 
002688172 





52 


No. 





TABLE XXIII.—SQUARES, CUBES, SQUARE ROOTS 


Squares. 


139129 
139876 
140625 
141376 
142129 
142884 
143641 





144400 
145161 
145924 
146689 
1477456 
148225 
148996 
149769 
150544 
151321 


152100 
152881 


154449 
155236 
156025 
156816 
157609 
158404 
159201 


160000 
160801 
161604 
162409 
163216 
164025 
164836 
165649 
166464 
167281 


168100 
168921 
169744 
170569 
171396 
172225 
173056 
173889 
174724 
175561 
176400 
177241 
178084 
178929 
179776 
180625 
181476 
182329 
183184 
184041 


184900 
185761 
186624 
187489 
188356 











153664 





Cubes. 


51895117 
52318624 
527343875 
53157376 
58582633 
54010152 
54439989 


54872000 
55306341 
55742968 


56181887 - 


56623104 
57066625 
57512456 
57960603 
58411072 
58863869 


59319000 
59776471 
60236288 
60698457 
61162984 
61629875 
62099136 
62570773 
63044792 
63521199 


64600000 
64481201 
64964808 
65450827 
65939264 
66480125 
66923416 
67419143 

7917312 
68417929 
68921000 
69426531 
69984528 
70444997 
70957944 
71473375 
71991296 
72511713 
73034632 
(3560059 


74088000 
74618461 
75151448 
75686967 
76225024 
76765625 
77308776 
77854483 
78402752 
(8953589 


79507000 
80062991 
80621568 
81182737 
81746504 





Square 
Roots. 


—_——. 


193132079 
19.38390796 
19.3649167 
19.3907194 
19.4164878 
19. 4422221 
19.4679223 


19. 4935887 
195192213 
195448203 
19.5703858 
195959179 
19 .6214169 
196468827 
19.6723156 
19.6977156 
19. 7280829 
19.7484177 
19.7737199 
19. 7989899 
19. 8242276 
19. 8494332 
19.8746069 
19. 8997487 
19. 9248588 
19.9499373 
19.9749844 


20.0000000 
20.0249844 
200499377 
20.0748599 
20.0997512 
20. 1246118 
20.1494417 
20.1742410 
20.1990099 
20.2237484 


20. 2484567 
20.2731349 
20.2977831 
208224014 
20.3469899 
20.8715488 
20.3960781 
20.4205779 
20.4450483 
20.4694895 


20.4939015 
20.5182845 
205426386 
205669638 
20 5912603 
20.6155281 
20.6397674 
20.6639783 
20.6881609 
20.7123152 


20. 7364414 
20.7605395 
20.7846097 
208086520 
20 .8326667 








Cube Roots. 


71984050 
7 .2048322 
72112479 
7.21 76522 
72240450 
7 .2804268 
723867972 


_ 7.2481565 


72495045 
72558415 
72621675 


” 72684824 


72747864 
72810794 
72873617 
7 .2936330 
7 .2998936 


78061436 
78123828 
738186114 
738248295 
7 .3810369 
@ 8872339 
7 .38434205 
78495966 
78557624 
7.8619178 
7 .3680630 
78741979 
78803227 
73864373 
7 .3925418 
7.38986363 
74047206 
74107950 
7 4168595 
74229142 


7 4289589 
7 .43849938 
7.4410189 
74470342 
74530399 
74590859 
74650223 
7.4709991 
74769664 
74829242 


74888724 
74948113 
75007406 
75066607 
75125715 
7 5184730 
7 5243652 
7 5302482 
7 5361221 
75419867 


75478423 
7 .5536888 
75595263 
7 5653548 
7.5711743 











Reciproeals. 


002680965 
002673797 
.002666667 
00265957. 
002652520 
002645503 
. 002638522 


002631579 
002624672 
002617801 
002610966 
002604167 
002597403 
002590674 
002583979 
002577320 
002570694 


.002564103 
002557545 
002551020 
002544529 
002538071 
.002531646 
002525253 
.002518892 
.002512563 
002506266 


.002500000 
002493766 
002487562 
.002481390 
002475248 
002469136 
002463054 
002457002 
002450980 
002444988 


002439024 
002433090 
.002427184 
002421308 
002415459 
0024096389 
002403846 
002398082 
002892344 
.002386635 


002380952 
.002375297 
002369668 
002364066 
002358491 
002352941 
.002847418 
002841920 
002336449 
.002331002 


002325581 
.002820186 
.002314815 
.002309469 
-002804147 


No. 








CUBE ROOTS, AND RECIPROCALS 


Squares, 


ee 


189225 
190096 
190969 

191844 
192721 


193600 
194481 
195364 
196249 
197136 
198025 
198916 
199809 
200704 
201601 
202500 
203401 
204304 
205209 
206116 
207025 
207936 
208849 
209764 
210681 


211600 
212521 
213444 
214369 
215296 
216225 
217156 
218089 
219024 
219961 
220900 
221841 
222784 
223729 
224676 
225625 
226576 
227529 
228484 
229441 


230400 
231361 
232324 
233289 
234256 
— 235225 
236196 
237169 
238144 
239121 


240100 
241081 
242064 
243049 
244036 
245025 
246016 





Cubes, 





82312875 
82881856 
83453453 
84027672 
84604519 


~ 85184000 


85766121 
86350888 
86938307 
87528384 
88121125 
88716536 
89314623 
89915392 
90518849 


91125000 
91733851 
92345408 
92959677 
93576664 
94196375 
94818816 
* 95443993 
96071912 
96702579 


97336000 

97972181 

98611128 

99252847 

99897344 
100544625 
101194696 
101847563 
102503232 
103161709 
103823000 
104487111 
105154048 
105823817 
106496424 
107171875 
107850176 
108531333 
109215352 
109902239 


110592000 
111284641 
111980168 
112678587 
113379904 
114084125 
114791256 
115501303 
116214272 
116930169 


, 117649000 


118370771 
119095488 
119823157 
120553784 
121287375 


122023936. 
——_ ————'['"—————e TEE Se eS Ta, 


Square 
Roots. 


20.8566536 
20. 8806130 
20. 9045450 
20. 9284495 
20 . 9523268 


20. 9761770 


21.1187121 
21.1423745 
21.1660105 
21.1896201 


21 .21382034 
21.2367606 
212602916 
21.2837967 
21 .8072758 
21.3307290 
21 .3541565 
213775583 
21.4009346 
214242853 


21 .4476106 
21.4709106 
21 .4941853 
21.5174348 
215406592 
215638587 
215870331 
216101828 
21.6333077 
21. 6564078 
21.6794834 
21. 7025344 
21.7255610 
21.7485632 
21.7715411 
21.7944947 
218174242 
218403297 
21 8632111 
21. 8860686 


21. 9089023 
21.9317122 
21.9544984 
21.9772610 


220227155 
22.0454077 
22.0680765 
22.0907220 
22.1133444 


22. 1359436 
22.1585198 
22.1810730 
22.2036033 
22 .2261108 
22..2485955 
22.2710575 


Cube Roots, 


7.576984) - 


75827865 
7.5885793 
7.5943633 
7.6001385 
6059049 
-6116626 
.6174116 
-6231519 
6288837 
-6346067 
6403213 
6460272 
6517247 
6574138 
-6630943 
.6687665 
6744803 
6800857 
6857328 
6913717 
.6970023 
- (026246 
7082388 
7138448 
7194426 
7250325 
7306141 
7361877 
74175382 
7473109 
7528606 
7584023 
7639361 
. 7694620 


7749801 
7804904 
7859928 
7914875 
7. 7969745 


IIS VWIG QV 


mF AF AF AT AF As AF TAT 


AEAZABAZ PAA Fag +7 


78133892 
78188456 
78242942 


78297353 
78351688 
7 .8405949 
78460134 
78514244 
78568281 
7 8622242 
78676130 
78729944 
7 8783684 


78837352 
78890946 
78944468 
78997917 
7.9051294 
7.9104599 
79157832 





53 





Reciprocals. 





.002298851 
002293578 
002288330 
002283105 
002277904 
-002272727 


002267574 


002262443 
002257336 
002252252 
002247191 
-002242152 
002237136 
002232148 


002227171 


002222222 
002217295 
002212389 
-002207506 
002202643 
002197802 
002192982 


002188184 


002183406 
002178649 
002173913 
002169197 
-002164502 
002159827 
002155172 
002150538 
002145923 
002141328 
002136752 
002182196 


.002127660 
002123142 
002118644 
.002114165 
002109705 
002105263 
.002100840 
.002096436 
-002092050 
.002087683 


002083333 
002079002 
002074689 
002070393 
002066116 
002061856 
.002057618 
002053388 
002049180 
002044990 


002040816 
.002036660 
002082520 
002028398 
002024291 
- 002020202 
-002016129 


54 





No. 








TABLE XXIII.—SQUARES, CUBES, SQUARE ROOTS 


Squares. 


247009 
248004 
249001 


250000 
251001 
252004 
253009 
254016 
255025 
256036 
257049 
258064 
259081 


260100 
261121 
262144 
263169 
264196 
265225 
266256 
267289 
268324 
269361 
270400 

“271441 
Q72484 
Q73529 
274576 
275625 
276676 
277729 
278784 
279841 
280900 
281961 
283024 
284089 
285156 
286225 
287296 
288369 
289444 
290524 


291600 
292681 
293764 
294849 
295936 
297025 
298116 
299209 
300304. 
801401 


302500 
803601 
804704 
805809 
806916 
808025 
809136 
810249 
311864 


Cubes. 


122763473 
123505992 
124251499 
125000000 
125751501 
126506008 
12726352 

128024064 
128787625 
129554216 
130323843 
131096512 
131872229 
132651000 
133432831 
134217728 
135005697 
135796744 
136590875 
137388096 
138188413 
138991832 
139798359 
140608000 
141420761 
142236648 
143055667 
143877824 
144703125 
145531576 
146363183 
147197952 
148035889 
148877000 
149721291 
150568768 
151419437 
152273304 
153130375 
153990656 
154854153 
155720872 
156590819 


* 157464000 


158340421 
159220088 
160103007 
160989184 
161878625 
162771336 
163667323 
164566592 
165469149 


166375000 
167284151 
168196608 
169112377 
170031464 
170953875 
171879616 
172808693 
178741112 


Square 
Roots. 


22. 2934968 
22.3159136 
22. 3383079 
223606798 
223830293 
22. 4053565 
224276615 
224499443 
224722051 
224944438 
225166605 
225388553 
225610283 


225831796 
22, 6053091 
22. 6274170 
22. 6495033 
22. 6715681 
22. 6936114 
22.7156334 
22.7376340 
22.7596134 
22. 7815715 
22. 8035085 
22. 8254244 
22.8473193 
22. 8691933 
22.8910463 
22. 9128785 
229346899 
22. 9564806 
22. 9782506 
23 0000000 


23 0217289 
23 04843872 
23 .0651252 
23. 0867928 
23. 1084400 
231800670 
23.1516738 
23 .1732605 
23. 1948270 
23.2163735 
23.2879001 
232594067 
23 . 2808935 
23 38023604 
23 .8238076 
23 8452351 
233666429 
23 .3880311 
23 4093998 
23 4807490 


23 .4520788 
23.4738892 
23 .4946802 
23 .5159520 
28 .5372046 
28 5584380 
235796522 
23 .6008474 
23. 6220236 


| 
Cube Roots. 


7.9210994 
79264085 
7.9317104 


79370053 
79422931 
9475739 
7.952847 
79581144 
7. 9633743 
7. 9686271 
79738731 
7. 9791122 
7. 9843444 
79895697 
79947883 
8.0000000 
8 .0052049 
8.0104032 
80155946 
8 .0207794 
8 0259574 
8.0311287 
80362935 


8.0414515 
8 .0466030 
8.0517479 
8. 0568862 
80620180 
80671432 
8 .0722620 
8.073743 
8 .0824800 
8.0875794 


80926723 
8 .097°7589 
8. 1028390 
8.1079128 
8.1129803 
8.1180414 
8. 1230962 
8.1281447 
8.1331870 
8. 1382230 


8.1432529 
8. 1482765 
8 .1532939 
8. 1583051 
8.1633102 
8.1683092 
81738020 
8. 1782888 
8. 1832695 
8.1882441 


8.1932127 
8.1981752 
8.2031319 
8. 2080825 
8.2180271 
8.2179657 
8 .2228985 
8 2278254 
8. 2327463 


Reciprocais. 


002012072 
. 002008032 
002004008 


002000000 
.001996008 
-001992032 
-001988072 
001984127 
.001980198 
.001976285 
001972387 
.001968504 
0019646387 
001960784 
-001956947 
-001958125 
-001949318 
-001945525 
.001941748 
-901937984 
.001934236 
-001930502 
-001926782 


001923077 
.001919386 
001915709 
.001912046 
.001908397 
.001904762 
.001901141 
001897533 
.001893939 
.001890359 


.001886792 
001883239 
001879699 
001876173 
001872659 
001869159 
-001865672 
001862197 
001858736 
001855288 


001851852 
001848429 
001845018 
.001841621 
001838235 
001834862 
-001831502 
001828154 
001824818 
001821494 


001818182 
.001814882 
001811594 
.001808318 
001805054 
-001801802 
001798561 
-001795332 
001792115 


Ss 


CUBE ROOTS, AND RECIPROCALS 


55 





No. 








Squares. Cubes. 
812481 174676879 
313600 175616000 
314721 176558481 
315844 177504328 
316969 178453547 
318096 179406144 
319225 | 180362125 
320356 | 181321496 
321489 | 182284263 
322624 | 183250432 
823761 184220000 
324900 185193000 
826041 186169411 
827184 187149248 
828329 188182517 
329476 189119224 
330625 190109375 
331776 191102976 
332929 192100033 
334084 193100552 
835241 194104529 
836400 | .195112000 
337561 196122941 
338724 197137368 
239889 198155287 
341056 199176704 
842225 200201625 
343396 2012380056 
344569 202262003 
345744 203297472 
346921 204336469 
348100 | 205379000 
349281 206425071 
850464 207474688 
351649 208527857 
352836 209584584 
854025 210644875 
355216 211708736 
356409 212776173 
357604 213847192 
358801 214921799 
360000 216000000 
361201 217081801 
362404. ‘218167208 
363609 219256227 
364816 220348864 
366025 221445125 
867236 222545016 
868449 223648543 
369664 224755712 
370881 225866529 
372100 226981000 
373321 228099131 
874544 229220928 
375769 230346397 
876996 }y «281475544 
878225 232608375 
879456 233744896 
380689 234885113 
881924 2360290382 
383161 237176659 
384400 238328000 


Square 
Roots. 





23.6431808 


23 .6643191 
23 .6854386 
23, 7065392 
23 7276210 
23. 7486842 
23. 7697286 
23 .7907545 
23.8117618 
23 8327506 
20.8587209 
23 8746728 
238956063 
23 .9165215 
28.9374184 
283. 9582971 
23.9791576 
24 .0000000 
24.0208243 
24.0416306 
24.0624188 
24.0831891 
24.1039416 
24.1246762 
24. 14538929 
24.1660919 
24.1867732 
24 2074369 
24. 2280829 
24.2487113 
24.2698222 


24.2899156 
24.3104916 
243310501 
24.3515913 
24.3721152 
243926218 
24.4131112 
24 4335834 
244540885 
244744765 
24.4948974 
24.5153013 
24 5856883 
245560583 
24.5764115 
245967478 
24.6170673 
24.6373 00 
24.6576560 
24.6779254 
24.6981781 
24.7184142 
247386338 
24.7588368 
24.7790284 
24.7991935 
24.8193473 
24 8394847 
24. 8596058 
24,8797106 
24.8997992 





Cube Roots. 


82376614 


8..2425706 
8.2474740 
8..2523715 
8. 2572633 
8.2621492 
8.2670204 
8.2719039 
8.2767726 
8..2816355 
82864928 
8.291344 
8.2961903 
8.3010304 
8.3058651 
8.3106941 
8.3155175 
83203353 
8.3251475 
83299542 
8.3347553 


83395509 
8.3443410 
8 .3491256 
8.3539047 
8 .3586784 
8 .3634466 
88682095 
8.3729668 
8.3777188 
83824653 


8 3872065 
8.3919423 
8 8966729 
8.4013981 
8.4061180 
84108326 
8.4155419 
8. 4202460 
84249448 
84296383 


8 4343267 
8 .4890098 
8.448687 
8. 4483605 
84530281 
8. 4576906 
8.4623479 
8.4670601 
8.4716471 
8.4762892 
8. 4809261 
8 .4855579 
8.4901848 
84948065 
8. 4994233 
8 .5040350 
8.5086417 
8.5182485 
8.5178403 
8.5224321 
8.5270189 


Reciprocals. 


001788909 


001785714 
001782531 
001779359 
001776199 
-001773050 
001769912 
001766784 
001768668 
001760563 
001757469 


001754386 
-0017518138 
-001 748252 
-001745201 
-001742160 
.0017391380 
001736111 
-001733102 
-001730104 
001727116 


001724138 
.001721170 
001718213 
001715266 
001712829 
001709402 
001706485 
-001703578 
001700680 
001697793 
001694915 
001692047 
-001689189 
001686341 
001683502 
-001680672 
-00167 7852 
.001675042 
001672241 
.001669449 


001666667 
.001663894 
.001661130 
.001658375 
001655629 
.001652893 
.001650165 
.001647446 
.001644737 
.001642036 


.001639344 
.001636661 
.001638987 
001631321 
-001628664 
.001626016 
-001623377 
.001620746 
-001618123 
.001615509 
.001612903 








56 
Square 

No. |Squares. Cubes. Boots: 
621 385641 239483061 24.9198716 
622 886884 240641848 24 .9399278 
623 888129 241804867 24 9599679 
624 389376 242970624 24.9799920 
625 390625 244140625 25 .0000000 
626 | 3891876 245314376 25.0199920 
627 | 3931239 246491883 25 0399681 
628 | 394384 2476738152 25 0599282 
62 395641 248858189 25 0798724 
630 396900 250047000 25 .0998008 
631 398161 251239591 25.1197184 
632 | 399424 252435968 25. 1896102 
633 400689 253636137 25 .1594913 
634 401956 254840104 25 .1793566 
635 403225 256047875 25. 1992063 
636 | 404496 257259456 - 25.2190404 
637 405769 258474853 25 . 2888589 
638 407044 259694072 25 .2586619 
639 408321 260917119 25 . 2784493 
640 409600 262144000 25 .2982213 
641 410881 263374721 25 .3179778 
642 412164 264609288 25 33877189 
643 413449 265847707 95 3574447 
644 414736 267089984 95 3771551 
645 416025 268336125 25 . 3968502 
646 417316 269586136 25 .4165301 
647 418609 270840023 25 .43861947 
648 419904 272097792 25 .4558441 
649 421201 273359449 25 4754784 
650 422500 274625000 25 .4950976 
651 423801 275894451 25.5147016 
652 425104 277167808 25 5342907 
653 426409 278445077 25 .5538647 
654 427716 279726264 25 5734237 
655 429025 281011375 25 5929678 
656 430336 282800416 25 .6124969 
657 431649 283593393 25 .6320112 
658 432964 284890312 25 .6515107 
659 434281 286191179 25 .6709953 
660 435600 287496000 25 .6904652 
661 436921 288804781 25 .7099203 
662 438244 290117528 25 7298607 
663 439569 291434247 25. 7487864 . 
664 440896 292754944 25 7681975 
665 442225 294079625 25 . 7875939 
666 443556 295408296 25 .8069758 
667 444889 296740963 25 .8263431 
668 446224 298077632 25 .8456960 
669 447561 299418309 25.8650343 
670 448900 800763000 25 . 8843582 
671 450241 3802111711 25 .9036677 
672 451584 303464448 25 . 9229628 
673 452929 804821217 25. 9422485 
674 454276 806182024 25.9615100 
675 455625 3807546875 25 .9807621 
676 456976 308915776 26 .0000000 
677 458329 310288733 26.0192237 
678 459684 311665752 26 .0384331 
679 461041 313046839 26 .0576284 
680 462400 - 814432000 26 0768096 
681 463761 315821241 26 .0959767 
682 465124 817214568 26.1151297 





TABLE XXIII.—SQUARES, CUBES, SQUARE ROOTS 











| 








Cube Roots. 


8.5316009 ° 


8.5361780 
8.5407501 
8.5453173 
8.5498797 
8.5544372 
8 .5589899 
856385377 
8.5680807 


8.5726189 
8.571523 


* 8.5816809 


8.5862047 
85907238 
8 .5952380 
8.5997476 
86042525 
8 .6087526 
8.6132480 
8.6177388 
8 .6222248 
8.6267063 
8.6311830 
8 .6356551 
8.6401226 
86445855 
8.6490437 
8.6534974 
8.6579465 


86623911 
8.6668310 
8.6712665 
8.6756974 
8.6801237 
8.6845456 
8.6889630 
8.6933759 
8.697843 
8.7021882 


. (0658777 


7416246 
7459846 


7503401 
7546913 
7590383 
7633809 
7677192 
7720532 
7763830 
7807084 
. 1850296 


8.7893466 - 


8 7936593 
8.7979679 
88022721 








Reciprocals, 


.001610306 
.001607717 
.0016051386 
001602564 
.001600000 
001597444 
-001594896 
.001592357 
-001589825 


.001587302 
.001584786 
.001582278 
.001579779 
-001577287 
-001574803 
-001572327 
-001569859 
.001567398 
001564945 


001562500 
001560062 
001557632 
.001555210 
-001552795 
- 001550388 
-001547988 
-001545595 
-001543210 
001540832 


.001538462 
.001536098 
001533742 
.001531394. 
901529052 
-001526718 
.001524390 
-001522070 
.001519757 
.001517451 


.001515152 
.001512859 
.001510574 
.001508296 
001506024 
001503759 
-001501502 
.001499250 
-001497006 
.001494768 


001492537 
-001490313 
.001488095 
001485884 
-001483680 
.001481481 
-001479290 
.001477105 
.001474926 
001472754 


001470588 
.001468429 
.001466276 








Squares. 


466489 
467856 
469225 
470596 
71969 
473344 
74724 
476100 
477481 
478864 
480249 
481636 
483025 
484416 
485809 
487204 
488601 
490000 
491401 
* 492804 
494209 
495616 
497025 
498436 
409849 
501264 
502681 


504100 
505521 
506944 
508369 
509796 
511225 
512656 
514089 
515524 
516961 


518400 
519841 
521284 
522729 
524176 
525625 
527076 
528529 
529984 
531441 


582900 
534361 
535824 
537289 
538756 
540225 
541696 
543169 
544644 
546121 
547600 
549081 
550564 
552049 
553536 





Cubes. 


318611987 
320013504 
321419125 
322828856 
324242703 
325660672 
327082769 
328509000 
329939371 
331373888 
332812557 
334255384 
335702375 
337158536 
338608873 
340068392 
341532099 
343000000 
344472101 
345948408 
347428927 
348913664 
350402625 
351895816 


358393243 


354894912 
356400829 
357911000 
359425431 
860944128 
362467097 
868994344 
365525875 
367061696 
363601813 
870146232 
371694959 


373248000 
874805361 
376367048 
377933067 
379503424 
881078125 
382657176 
384240583 
885828352 
387420489 


389017000 
890617891 
392223168 
395832837 
395446904 
397065375 
398688256 
400315553 
401947272 


” 403583419 


405224000 
406869021 
408518488 
410172407 
411830784 








Square 
Roots. 





26. 1342687 
26. 1533937 
26. 1725047 
26. 1916017 
26 2106848 
26 2297541 
26. 2488095 
26. 2678511 
26 2868789 
26 3058929 
26 3248932 
26. 3438797 
26. 3628527 
263818119 
26 4007576 
26.4196896 
26. 4386081 
264575131 
26.4764046 
26. 4952826 
265141472 
26 5320983 
265518361 
26 5706605 
26 5894716 
26 . 6082694 
26 6270539 
26 6458252 
26. 6645833 
26 . 6833281 
26. 7020598 
26.7207 

26. 7394839 
26. 7581763 
26. 7768557 
26 .'7955220 
26 8141754 
26 8328157 
26 8514432 
26 8700577 
26 8886593 
26 9072481 
26 9258240 
26 9443872 
26 . 9629375 
26. 9814751 
270000000 


27 0185122 
270370117 
27 .0554985 
27 .0739727 
27 0924344 
27.1108834 
27 .1293199 
271477439 
27.1661554 
27 1845544 
27 2029410 
27 2213152 
27 2396769 

72580263 
27 2763634 





CUBE ROOTS, AND RECIPROCALS 


Cube Roots. 





8 .8065722 
8.8108681 
8.8151598 
8.8194474 
8 .8237307 
8.8280099 
88322850 


8 .8365559 
8 .8408227 
8.8450854 
8.8493440 
8.8535985 
8.8578489 
8.8620952 
8.8663375 
8.8705757 
8.8748099 


8.8790400 
8.8832661 
8.8874882 
8.8917063 
8.8959204 
8. 9001304 
8. 9043366 
8. 9085387 
8.9127369 
8.9169311 
8.9211214 
8.92538078 
8.9294902 
8, 9336687 
8. 9378483 
89420140 
8.9461809 
8. 9503438 
89545029 
8.9586581 
89628095 
8. 9669570 
8.9711007 
89752406 
8.9793766 
8.9835089 
8.9876373 
8.9917620 
8. 9958829 
9 .0000000 


90041134 
9 0082229 
9 0123288 
9.0164809 
9 0205293 
9 0246239 
9 0287149 
9 .0328021 
9 .0368857 
9.0409655 


9 .0450419 
9.0491142 
9.0531831 
9 .0572482 
9 .0613098 


Reciprocals. 


.001464129 
001461988 
001459854 
001457726 
001455604 
001453488 
.001451379 


001449275 
.001447178 
.001445087 
.001443001 
.001440922 
.001438849 
.001436782 
.001434720 
.001482665 
.001480615 


.001428571 
.0014265384 
.001424501 
.001422475 
001420455 
.001418440 
.001416431 
.001414427 
.001412429 
001410437 


.001408451 
001406470 
.001404494 
.001402525 
.001400560 
.001398601 
001396648 
.001394700 
001392758 
.001390821 


.001388889 
,V013886963 
001885042 
.001383126 
.0013881215 
.001379310 
.001377410 
.001375516 
001373626 
001371742 


001369863 
.0013867989 
.001866120 
.001364256 
001362398 
001360544 
.001358696 
001356852 
.001855014 
.001353180 


.001351351 
001349528 
.001347709 
.001345895 
.001344086 





58 











Squares. 


555025 
556516 
558009 
559504 
561001 
562500 
564001 
565504 
567009 
568516 
570025 
571536 
573049 
574564 
576081 


577600 
579121 
580644 
582169 
583696 
585225 
586756 
588289 
589824 
591361 


592900 
594441 
595984 
597529 
599076 
600625 
602176 
603729 
605284 
606841 


608400 
609961 
611524 
613089 
614656 
616225 
617796 
619369 
620944 
622521 


624100 
625681 
627264 
628849 
630436 
632025 
633616 
635209 
636804 
638401 


640000 
641601 
643204 
644809 
646416 
648025. 
649636 








Cubes. 





413493625 
415160936 
416832723 
418508992 
420189749 
421875000 
423564751 
425259008 
426957777 
428661064 
430868875 
432081216 
433798093 
435519512 
437245479 


438976000 
440711081 
442450728 
444194947 
445943744 
447697125 
449455096 
451217663 
452984832 
454756609 
456533000 
458314011 
460099648 
461889917 
463684824 
465484375 
467288576 
469097433 
470910952 
472729139 
4774552000 
476379541 
478211768 
480048687 
481890304 
483736625 
485587656 
487443403 
489303872 
491169069 


493039000 
494913671 
496793088 
498677257 
500566184 
502459875 
504358336 
506261573 
508169592 
510082399 


512000000 
513922401 
515849608 
517781627 
519718464 
521660125 
528606616 








Square 
Roots. 


272946881 
27 .3130006 
27 38313007 
27 8495887 
278678644 
27 3861279 
27. 4043792 

7 4226184 
27 4408455 

7 4590604 
27 4772032 
27 4954542” 
275186830 
27 5317998 
275499546 


275680975 
27 5862284 
27 .6043475 

7 6224546 
27 6405499 
27 6586334 
27 .6767050 
27 6947648 
RT .7128129 
277308492 
27 . 7488739 
27. 7668868 
27 7848880 
27 8028775 
7 8208555 
7 .8388218 
7 8567766 
27 .8747197 
7 8926514 
79105715 


27 9284801 
7 946877 
27°. 9642629 
27 .9821372 
28 .0000000 
28 .0178515 
28 0356915 
28 0535203 
280713377 
28.0891438 
28. 1069386 
28 . 12477222 
28. 1424946 
28. 1602557 
28. 1780056 
28.1957444 
28 2134720 
28 2311884 
28 . 2488938 
28 2665881 


28 .2842712 
28 3019434 
28 3196045 
28 3372546 
28 3548938 
28 8725219 
28. 3901391 








TABLE XXIII.—SQUARES, CUBES, SQUARE ROOTS 


Cube Roots. | Reciprocals. 


9.065367 
9 0694220 
9.0734726 
9.0775197 
90815631 
9 .0856030 
9 0896392 


. 1218010 


. 1258053 
1298061 
. 1388034 
1877971 
1417874 
1457742 
1497576 
1537375 
1577189 
1616869 


1656565 
1696225 
1735852 
1775445 
1815003 
1854527 
1894018 
. 1933474 
9.1972897 
9 .2012286 


9.2051641 
9. 2090962 
92130250 
9.2169505 
9 2208726 
9.2247914 
9 2287068 
92326189 
9.236527 
9. 24043383 


9 2443855 
9 2482344 
9 .2521800 
9 .2560224 
9.2599114 
9.2637978 
9 .2676798 
92715592 
9 2754852 
9 .2798081 


9.288177 
9 .2870440 
9 .2909072 
9.294671 
9 2986239 
9. 8024775 
93063278 





-001842282 
001340483 
.001338688 
.001836898 
001835113 


001333333 
.001881558 
001829787 
.001328021 
.001826260 
.001824503 
001822751 
001821004 
.001319261 
.001317523 


.001315789 
001314060 
.001812336 
.001310616 
.001308901 
.001307190 
.001305483 
.001808781 
.001302083 
001300390 


001298701 
-001297017 
001295337 
.001298661 
.001291990 
001290823 
.001288660 © 
.001287001 
001285347 
.001288697 


001282051 
001280410 
001278772 
001277139 
001275510 
001273885 
001272265 
001270648 
001269036 
001267427 
001265823 
001264223 
001262626 
001261034 
0012594405 
001257862 
001256281 
001254705 
001253133 
001251564 
001250000 
001248439 
001246883 
001245330 
001243781 
001242236 
001240695 





CUBE ROOTS, AND RECIPROCALS 


Squares, 


651249 
652864 
654481 
656100 
657721 
659344 
660969 
662596 
664225 
665856 
667489 
669124 
670761 
672400 
674041 
675684 
677329 
678976 
680625 
682276 
683929 
685584 
687241 
688900 
690561 
692224 
693889 
695556 
697225 
698896 
700569 
702244 
703921 
705600 
707281 
708964 
710649 
712336 
714025 
715716 
"17409 
719104 
720801 


722500 
724201 
725904 
727609 
729316 
731025 
732736 
734449 
736164 
737881 


739600 
741321 
748044 
744769 
746496 
748225 
749956 
751689 
758424 


Cubes. 


525557943 
527514112 
529475129 
531441000 
583411731 
535387328 


537367797 _ 


539353144 
541343375 
543338496 
545338513 
547343432 
549353259 
551368000 
553387661 
555412248 
557441767 
559476224 
561515625 
563559976 
565609283 
567663552 
569722789 


571787000 
573856191 
575930368 
578009537 
580093704 
582182875 
584277056 
586376253 
588480472 
590589719 


592704000 
594823321 
596947688 
599077107 
601211584 
603351125 
605495736 
607645423 
609800192 
611960049 


614125000 
616295051 
618470208 
620650477 
622835864 
625026375 
627222016 
629422793 
631628712 
633839779 


636056000 
638277381 
640503928 


"642735647 


644972544 
647214625 
649461896 
651714363 
653972032 


Square 
Roots. 





28 4077454 


28 4253408 
28 4429253 


28 .4604989 
28 .4780617 
28 4956137 
285181549 
28 .53806852 
28 5482048 
28 .5657137 
28 5882119 
28 6006993 
28 .6181760 


28 .6356421 
28 6530976 
28 6705424 
28 .6879766 
28 . 7054002 
28 . 7228132 
28 7402157 
28.75 76077 
28. 7749891 
28 . 7923601 


28 8097206 
28 8270706 
28 8444102 
28 .8617394 
28 .8790582 
288963666 
28 . 9136646 
28 . 9309523 
28 , 9482297 
28. 9654967 


28 . 9827535 
290000000 
29.0172363 
29. 0344623 
29.0516781 
29 0688837 
290860791 
29. 1032644 
29 . 1204396 
29. 1376046 


29, 1547595 
29.1719043 
29. 1890390 
29. 2061637 
29 2282784 
29 2403830 
29.2574 777 
29 2745623 
292916370 
293087018 


29. 3257566 
293428015 
29 38598365 
293768616 
29 3988769 
29 4108823 
294278779 
294448637 
29 4618397 








9.3101750 
9.3140190 
9.3178599 


98216975 
9 .3255320 
9 .82938634 
9.3331916 
9.3370167 
9.3408386 
93446575 
93484731 
9 38522857 
9 8560952 


9 .3599016 
9.3637049 
9.3675051 
9 .3718022 
98750963 
9. 3788873 
9 8826752 
9 .3864600 
9 3902419 
9 .3940206 


9.3977964 
9.4015691 
9 4053387 
9.4091054 
9.4128690 
94166297 
9 4203873 
9.4241420 
9. 4278936 
94316423 


9 4353880 
9.4391307 
9. 4428704 
9. 4466072 
9 .4503410 
9.4540719 
9.4577999 
94615249 
9. 4652470 
94689661 


9 4726824 
94768957 
9.4801061 
9 4838136 
94875182 
9.4912200 
9.4949188 
9.4986147 
95023078 
9.5059980 


9.5096854 
9.5133699 
9.5170515 
9.5207303 
9 .5244063 
9 .5280794 
9.5317497 
9 .5354172 
9.5390818 


59 





Cube Roots. | Reciprocals. 





001239157 
001237624 
001236094 


001234568 
.001233046 
001231527 
.001230012 
.001228501 
001226994 
001225490 
001223990 
001222494 
.001221001 


.001219512 
.001218027 
001216545 
001215067 
0012138592 
.001212121 
001210654 
.001209190 
001207729 
.001206273 


001204819 
001203369 
001201923 
001200480 
.001199041 
.001197605 
.001196172 
.001194743 
.001198317 
.001191895 


.001190476 
.001189061 
.001187648 
001186240 
001184834 
.001183432 
.001182033 
.001180638 
001179245 
.00117'7856 


.001176471 
001175088 
.001173709 
.001172333 
001170960 
.001169591 
001168224 
.001166861 
001165501 
-001164144 


001162791 
.001161440 
.001160093 
001158749 
001157407 
.001156069 
001154734 
001153403 
001152074 





60 





TABLE XXIII.—SQUARES, CUBES, SQUARE ROOTS 





No. Squares. 
869 755161 
870 756900 
871 758641 
872 760384 
873 762129 
874 763876 
875 765625 
87 767376 
877 769129 
878 770884 
879 772641 
880 774400 
881 776161 
882 V77924 
883 V79689 
884 781456 
885 (83225 
886 784996 
887 . (86769 
888 788544 
889 790321 
890 792100 
891 793881 
892 795664 
893 797449 
894. 799236 
895 801025 
896 802816 
897 804609 
898 806404 
899 808201 
900 810000 
901 811801 
902 813604 
903 815409 
904 817216 
905 819025 
906 820836 
907 822649 
908 824464 
909 826281 
910 828100 
911 829921 
912 831744 
913 833569 
914 835396 
915 837225 
916 839056 
917 840889 
918 842724 
919 844561 
920 846400 
921 848241 
922 850084 
923 851929 
924 853877 

925 855625 
926 857476 
927 859329 
928 861184 
929 863041 
930 864900 





Cubes. 


656234909 


658503000 
660776311 
668054848 
665338617 
667627624 
669921875 
672221376 
674526133 
676836152 
679151439 


681472000 
683797841 
686128968 
688465387 
690807104 
693154125 
695506456 
697864103 
700227072 
702595369 


704969000 
(07347971 
709732288 
712121957 
714516984 
716917375 
719323136 
721734273 
724150792 
726572699 
729000000 
731482701 
733870808 
736314827 
738763264 
741217625 
743677416 
746142643 
748613312 
751089429 


753571000 
756058031 


758550528 


761048497 
763551 944 
766060875 
7168575296 
771095213 
773620632 
776151559 


778688000 
781229961 
783777448 
786330467 
788889024 
791453125 
794022776 
796597983 
799178752 
801765089 
804357000 





Square 
Roots. 


29.4788059 


29 4957624 
295127091 
295296461 
295465734 
29 5634910 
295803989 
29 5972972 
296141858 
29 .6310648 
296479342 


29.6647939 
29.6816442 
29 6984848 
29.7153159 
29 .7821875 
29.7489496 
29. 7657521 
29 . 7825452 
29 ..7998289 
29.8161030 
29 .8328678 
29 8496231 
298663690 
29.8881056 
298998328 
299165506 
299382591 
299499583 
29.9666481 
29. 9833287 


30.0000000 
30.0166620 
30 ..0333148 
30.0499584 
30.0665928 
30.0832179 
30.0998339 
30.1164407 
30. 1330383 
380. 1496269 


30. 1662063 
30. 1827765 
30.1993377 
30. 2158899 
30 2324329 
30 .2489669 
80.2654919 
30. 2820079 
30 .2985148 
80.3150128 


30.3315018 
30.3479818 
30 ..3644529 
30.3809151 
80.39738683 
30.4138127 
30. 4302481 
30. 4466747 
30. 4630924 
380. 4795013 
30.4959014 








9.5427437 


9 .5464027 
9 .5500589 
9 .5537123 
9 .5573630 
9.5610108 
95646559 
95682982 
95719377 
9.5755745 
9.5792085 


9 .5828397 
9 5864682 
95900939 
9.5937169 
9 5973373 
9.6009548 
9.6045696 
9.6081817 


_9.6117911 


9.615397 


9.6190017 
9 .6226030 
9.6262016 
96297975 
9. 6333907 
9.6369812 
9.6405690 
96441542 
9.647367 
9.6518166 


9 .6548938 
9 6584684 
9 6620403 
96656096 
96691762 
9.6727403 
9.6763017 
96798604 
96834166 
9.6869701 


9.6905211 
9.6940694 
9.6976151 
9.7011583 
9.7046989 
9 .'7082369 
7117728 


. 7188354 
. 7223681 


7258883 
7294109 
7329309 
7364484 
73996384 
7484758 
7469857 
7504930 
. 1539979 
9.'7575002 
9.7610001 


7158051" 


: 





Cube Roots. | Reciprocals. 


001150748 


.001149425 
.001148106 
.001146789 
.001145475 
:001144165 
.001142857 
.001141553 
.001140251 
.001188952 
.001137656 


.001136364 
.001135074 
.001133787 
.001182503 
0011381222 
.001129944 
.001128668 
.001127396 
.001126126 
001124859 


001123596 
001122334 
001121076 
001119821 
001118568 
001117318 
001116071 
001114827 
001113586 
001112347 
001111111 
001109878 
001108647 
001107420 
001106195 
001104972 
001103753 
001102536 
001101322 
001100110 


.001098901 
.001097695 
.001096491 
001095290 
.001094092 
001092896 
.001091703 
.001090513 
001089325 
001088139 


_ .001086957 
.001085776 
.001084599 
.001083423 
.001082251 
.001081081 
001079914 
001078749 
001077586 
.001076426 
001075269 


CUBE ROOTS, AND RECIPROCALS 


61. 





No. Squares. 








931 866761 
932 868624 
933 870489 
934 872356 
935 74225 
936 876096 
937 877969 
938 879844 
939 881721 
940 883600 
941 885481 
942 887364 
943 889249 
944 891136 
945 893025 
946 894916 
947 896809 
948 898704 
949 900601 
950 902500 
951 904401 
952 906304 
953 908209 
954 910116 
955 912025 
956 913936 
957 915849 
958 917764 
959 919681 
960 921600 
961 923521 
962 925444 

- 963 927369 
964 929296 
965 - | 931225 
966 933156 
967 935089 
968 37024 
969 938961 
970 940900 
971 942841 
972 944784 
97% 946729 
974 948676 
975 950625 
97 952576 
977 954529 
978 956484 
979 958441 
980 960400 
981 962361 
982 964324 
983 966289 
984 968256 
985 970225 
986 972196 
987 974169 
988 976144 
989 978121 

- 990 980100 
991 982081 
992 | 984064 


Cubes. 


806954491 
809557568 
812166237 
814780504 
817400375 


* 820025856 


822656953 
825293672 
827936019 


830584000 
833237621 
835896888 
838561807 
841232384 
8438908625 
846590536 
849278123 
8519713892 
854670349 


857375000 
860085351 
862801408 
865523177 
868250664 

10983875 


873722816 


876467493 
879217912 
881974079 
884736000 
887503681 
890277128 
893056347 
895841344 
898632125 
901428696 
904231063 
907039232 
909853209 
912673000 
915498611 
918330048 
921167317 
924010424 
926859375 
929714176 
932574833 
935441352 
938313739 


941192000 
944076141 
946966168 
949862087 
952763904 
955671625 
958585256 
951504803 
964430272 
967361669 


70299000 
973242271 
976191488 





Square 
Roots. 


30 .5122926 
30 .5286750 
80 .5450487 
80.5614136 
80.5777697 
80.5941171 
80. 6104557 
30.6267857 
380.6431069 


80.6594194 
30.67572383 
380.6920185 
30.7088051 
80.7245830 
380 .'7408523 
380.7571130 
30.7733651 
30.7896086 
30. 8058436 


808220700 
80 .8382879 
30 .8544972 
30 .8706981 
380. 8868904 
380. 9030743 
380.9192497 
30. 9354166 
80.9515751 
30. 9677251 
380. 9838668 
31.0000000 
31.0161248 
31 .0322413 
81.0483494 


erg 


- 81.0644491 


31 .0805405 
31 .0966236 
31. 1126984 
31. 1287648 


81. 1448230 
81. 1608729 
81.1769145 
81.1929479 
31 .2089731 
31 ..2249900 
31.2409987 
31. 2569992 
31.2729915 
31. 2889757 


31.3049517 
31.8209195 
81 .3368792 
81 .3528308 
381.3687743 
31.3847097 
31 .4006369 
31.4165561 
31 .43824673 
31 .4488704 


381 .4642654 
31 .4801525 


. 81.4960315 





9 .7644974 
9.7679922 
9.7'714845 
9.749743 
9.784616 
9.7819466 
9 .7854288 
9.7889087 
9.'7923861 
9 .'7958611 
9.7993886 
9. 8028036 
9.8062711 
9.8097362 
9.8131989 
9.8166591 
9.8201169 
9 .8285723 
9 8270252 
9.8804757 
9.8339238 
9 .8373695 
9.8408127 
98442536 
9.8476920 
9.8511280 
9.8545617 
9 8579929 
98614218 
9 8648483 
9. 8682724 
9.8716941 
9.8751135 
9.8785305 
9.8819451 
9.8853574 
9.8887673 
9.8921749 
9.8955801 


9.8989830 
9. 9028885 
9.9057817 
9.9091776 
9.9125712 
9. 9159624 
9.9193513 
9 9227879 
9 9261222 
9. 9295042 


9. 9328839 
9. 9362613 
9. 9396363 
9.9430092 
9.9468797 
9.9497479 
9 95311388 
9.956475 
9. 9598389 
9.9631981 


9. 9665549 
9.9699095 
9.9732619 





Cube Roots. | Reciprocails. 





.001074114 
001072961 
.001071811 
.001070664 
.001069519 
-001068376 
-001067236 
-001066098 
001064963 


. 001063830 
-001062699 
.001061571 
-001060445 
001059322 
001058201 
001057082 
.001055966 
.001054852 
-001053741 


.001052632 
-001051525 
-001050420 
.001049318 
001048218 
.001047120 
.001046025 
001044932 
-001043841 
-001042753 


001041667 
.001040583 
-001089501 
001088422 
.001037344 
.001086269 
.001035197 
001034126 
.001033058 
.001081992 


.001030928 
001029866 
001028807 
-001027749 
.001026694 
001025641 
.001024590 
001028541 
001022495 
-001021450 


001020408 
.001019368 
-001018330 
.001017294. 
.001016260 
.001015228 
.001014199 
.001013171 
.001012146 
.001011122 


.001010101 
001009082 
001008065 


(a —— 


62 





me | | | 





TABLE XXIII.—SQUARES, CUBES, SQUARE ROOTS 


986049 
988036 
990025 
992016 
994009 
996004 
998001 
1000000 


1002001 
1004004 
1006009 
1008016 
1010025 
1012036 
1014049 
1016064 
1018081 
1020100 


1022121 
1024144 
1026169 
1028196 
1030225 
1032256 
1034289 
1036324 
1038361 
1040400 


1042441 
1044484 
1046529 
1048576 
1050625 
1052676 
1054729 
1056784 
1058841 
1060900 


1062961 


1081600 


1083681 
1085764 
1087849 
1089936 
1092025 
1094116 
1096209 
1098304 
1100401 
1102500 


1104601 
1106704 
1108809 
1110916" 


979146657 
982107784 
985074875 
988047936 
991026973 
994011992 
997002999 
1000000000 


1003003001 
1006012008 
1009027027 
1012048064 
1015075125 
1018108216 
1021147343 
1024192512 
1027243729 
1030301000 


1033364331 


1036433728 © 


1039509197 
1042590744 
1045678375 
1048772096 
1051871913 
1054977832 
1058089859 
1061208000 


1064332261 | 


1067462648 
1070599167 
1073741824 
1076890625 
1080045576 
1083206683 
1086373952 
1089547389 
1092727000 


1095912791 
1099104768 
1102302937 
1105507304 
1108717875 
1111934656 
1115157653 
1118386872 
1121622319 
1124864000 


1128111921 
1131366088 
1184626507 
1137893184 
1141166125 
1144445336 
1147730823 
1151022592 
1154320649 
1157625000 


1160935651 
1164252608 
1167575877 
1170905464 


Square 
Roots. 


31.5119025 
31 .5277655 
31.5436206 
31.5594677 
81.5753068 
31.5911380 
31 .6069613 
31. 6227766 


31.6385840 
31. 6543836 
31.6701752 
81. 6859590 
31.7017349 
31.'7175030 
31.7332633 
31.7490157 
31.'7647603 
31.7804972 
31. 7962262 
31.8119474 
318276609 
31.8433666 
31.8590646 
31.8747549 
31.8904374 
31. 9061123 
31.9217794 
31.9374388 


31.9530906 
31.9687347 
31. 9843712 
82.0000000 
82.0156212 
82..0312348 
382.0468407 
82 .0624391 
82.0780298 
32.0936131 


32.1091887 
82.1247568 
32.1403173 
82.1558704 
82. 1714159 
82.1869539 
32. 2024844 
82.2180074 
82. 2335229 
82.2490310 


32 ..2645316 
32. 2800248 
82. 2955105 
82.3109888 
82 .3264598 
82. 3419233 
32.3573794 
82 .3728281 
823882695 
32. 4037035 


82.4191301 
82.4345495 
32.4499615 
82 .4653662 


Cube Roots. | Reciprocals. 


9.9766120 
9.9799599 
9 .9833055 
9.9866488 
9.9899900 
9. 9933289 
9.9966656 
10.0000000 


10.0038322 
10. 0066622 
10.0099899 
10.0133155 
10.0166389 
10.0199601 
10.0232791 
10.0265958 
10.0299104 
10.0332228 
10..0365330 
10.0398410 
10.0431469 
10.0464506 
10.0497521 
10.0530514 
10.0563485 
10..0596435 
10.0629364 
10.0662271 


10.0695156 
10.0728020 
10.0760863 
10.0793684 
10.0826484 
10.0859262 
10.0892019 
10.0924755 
10.0957469 
10.0990163 


10. 1022835 
10. 1055487 
-10.1088117 
10.1120726 
10.1153314 
10.1185882 
10 1218428 
10 .1250953 
10 1283457 
10.1815941 


10. 1348403 
-10. 1880845 
10. 1413266 
10. 1445667 
10.1478047 
10. 1510406 
10.1542744 


10. 1575062 


10.1607359 
10. 1639636 


10.1671893 
10. 1704129 
10.1736344 
10.1768539 


.001007049 
001006036 
001005025 
001004016 
:001003009 
001002004 
"001001001 
-001000000 


.0009990010 
-0009980040 
.0009970090 
.0009960159 
. 0009950249 
.0009940358 
.0009930487 
-0009920635 
.0009910803 
-0009900990 


.0009891197 
0009881423 
.0009871668 
,0009861933 
.0009852217 
-0009842520 
0009832842 
0009823183 
-0009813543 
0009803922 


0009794319 
"0009784736 
0009775171 
0009765625, 
0009756098 
-0009746589 
0009737098 
0009727626 
'0009718173 
0009708738 


.0009699321 
.0009689922 
.0009680542 
.0009671180 
.0009661836 
.0009652510 - 
.0009643202 
.0009633911 
0009624639 
0009615385 


.0009606148 
-0009596929 
0009587738 
.0009578544 
-0009569378 
.0009560229 
-0009551098 
.0009541985 
0009532888 
.0009523810 


.0009514748 
-0009505703 
.0009496676 
-0009487666 











TABLE XXIV.—LOGARITHMS 





















































































































































OF NUMBERS 

















63 


No, 100 L, 000.] [No. 109 L. 040. 
N.| 0 1 2 8 4 5 6 7 8 9 | Diff. 
100 | 000000 4 | 0868 | 1301 | 1734 || 2166 | 2598 | 3029 | 3461 | 3891 432 
1 4321 | 4751 | 5181 | 5609 | 6038 || 6466 | 6894 | 7321 | 7748 | 8174 | 428 
2 8600 | 9026 | 9451 | 9876 2 en 
0300 || 0724 | 1147 | 1570 | 1998 | 2415 | 424 
3 | 012837 | 3259 | 3680 | 4100 | 4521 || 4940 | 5360 | 5779 | 6197 | 6616 | 420 
4 7033 | 7451 | 7868 | 8284 | 8700 || 9116 | 95382 | 9947 —_. 
a. 0361 | O775 | 416 
5 | 021189 | 1603 | 2016 | 2428 | 2841 || 3252 | 3664 | 4075 | 4486 | 4896 | 412 
6 5306 | 5715 | 6125 | 6538 | 6942 || 73850 | 7757 | 8164 | 8571 | 8978 | 408 
7 9384 | 9789 ~ —s pairs. oo) 
——— 0195 | 0600 | 1004 || 1408 | 1812 | 2216 | 2619 | 3021 404 
8 | 033424 | 3826 | 4227 | 4628 | 5029 || 5430 | 5830 | 6230 | 6629 | 7028 | 400 
9 7426 | 7825 | 8223 | 8620 | 9017 || 9414 | 9811 —— _ 
04 : 207 | 0602 | 0998 | 397 
PROPORTIONAL PARTS. 
Diff, 1 2 3 4 5 6 v4 8 9 
434 | 43.4 | 86.8 1oOheuhe LiosOe= 2UGL0 260-4.) S03,8n 34772517 8O086 
433. | 43.3 | 86.6 129.9 TiS-cal eel ale POO se al Ose 346.4 | 389.7 
452 | 43.2.1°-86.4 129.6 72-8. 216.0.) 259.2) 302.4). 845s6rP38sa8 
431 43.1 86.2 129.3 U2 se Ope RosiGa Ol wey O44 soa oomeD 
430 | 483.0] 86.0 129,02) 172-0.;), 215.0 4) 258°05| - 301.04, 34420: 38720 
429- | 42.9} 85.8 128.7 NG 214 ball bin 300.3 843.2 | 386.1 
428 | 42.8! 85.6 128.4 171.2 | 214.0 | 256.8 299.6 | 342.4 | 385.2 
42% | 42.7) 85.4 128.1 170.8 | 218.5.| 256.2 | 298.9 | 3841.6 | 384.38 
426. | 42.6 | 85:2 AEB m4 BIS20 4) 255764) 298-251 84078) |) Sabed 
425 | 42.5.) 85.0 a Peet L7OcOnl) Slerae) 2op.08) 29 Sel 34050") 38275 
424 | 42.4 84.8 127.2 169.6 | 212.0 | 254.4 | 296.8] 339.2 | 381.6 
473. | 42°34) 84.6 126.9 1692 ileal 253280 296" 1 338.4 | 3880.7 
AV2en| 43.05) 84.4 126.6 168-8 | 211.0 | 253.2) 295.4) 837.6.) 379.8 
421 42.1 84.2 126.3 1684+ 210855) 2522673) 294%) - 33628: | 34829 
420 | 42.0, 84.0 126.0 168.0 | 210.0 | 252.0 | 294.0} 836.0 | 378.0 
419 | 41.9] 83.8 abease AG Ga LOOwael 251 45° 293738) | 338592) sume 
418 | 41.8] 88.6 125.4 167.2 | 209.0 | 250.8 | 292.6 | 334.4 | 376.2 
AIT, | 41e%') 8a4 125.1 16628.) 208.5.) 250921) 291.9") 338.6)| 37573 
416 | 41.6 | 83.2 124.8 | 166.4] 208.0] 249.6 | 291.2 | 3382.8 | 374.4 
415 | 41.5) 83.0 124.5 166.0 | 207.5; 249.0; 290.5 | 3832.0 | 3738.5 
414. | 41.4 | 82:8 124.2 165.6 | 207.0 | 248.4] 289.8 | 3831.2 | 372.6 
413. | 41.3) 82.6 123.9 165.2 | 206.5) 247.8) 289.1 830.4 | 371.7 
Ave) (412 Dob (824 123.6 | 164.8 | 206.0 722) 288.4") 82956. |) 387078 
411 41.1 82.2 (23:50) 164.4) 205-541 246°6 | 287.7 |. 328.8 |) 86979 
410 | 41.0-) 82.0 123.0 164.0 | 205.0 | 246.0] 287.0 | 3828.0 | 369.0 
409 | 40.9} 81.8 1225% 163.6 | 204.5 | 245.4 | 286.3} 327.2 | 368.1 
408 | 40.8} 81.6 122.4 163.2 | 204.0 | 244.8] 285.6} 3826.4 | 367.2 
407 | 40.7 |. 81.4 122.1 162.8 | 208.5 | 244.2] 284.9 | 325.6 | 366.3 
406 | 40.6 | 81.2 121.8 162.4 | 203.0 | 2486) 284.2 | 324.8 | 365.4 
405 | 40.5 | - 81.0 121.5 | 162.0; 202.5 | 248.0:) 283.51 3824:0 | 364.5 
404 | 40.4] 80.8 12199) 16426 202.0) 242%4*) 282-82). 323-2 | 363.6 
403 | 40.3] 80.6 120590) 1G1e2 3) 20125)| 241.81) 282.15) 8224 | 3627 
402 | 40.2! 80.4 120760) 16028%)) 201207) 2412+) 281.45) Beri6: > 36188 
401 40.1 80.2 %120.3 160.4 | 2005} 2406) 280.7] 320.8 | 360.9 
400 | 40.0 | 800 120.0 | 160.0] 200.0} 240.0 | 280.0) 820.0 | 360.0 
399 | 39.9 19.8 119.7 | 159.6 199.5 | 2389.4 | 279.38 |} 319.2 | 359.1 
398 | 39.8 | 79.6 119.4 | 159.2 199.0 | 288.8} 278.6 | 318.4 | 358.2 
897 | 89.7) 94 119.1 158.8 | 198:5 | 238.2) 277.9 | 317.6 | 357.38 
396 | 39.6 | 79.2 118.8 | 158:4]) 198.0} 237.6 | 277.21 3816.8 | 356.4 
395 | 39.5 | 79.0 118.5 | 158.0| 197.5! 287.01 276.5! 316.0 | 355.5 





64 TABLE XXIV.—LOGARITHMS OF NUMBERS 


No. 110 L, 041.] [No. 119 L. 078. 





NG 0 -| 1 2 8 oa 5 6 Cl Be «| 9 | Diff. 


—_ (| | | |  — ———__ | |  ———————_- — | | ——_——_ 


110 | 041393 | 1787 | 2182 | 2576 | 2969 || 2362 | 3755 | 4148 | 4540 | 4932 | 398 

5823 | 5714 | 6105 | 6495 | 6885 || 7275 | 7664 | 8053 | 8442 | 8830 | 390 

9218 | 9606 | 9993.|——— ———_ | ——_ 

0380. | 0766 || 1153 | 1538 | 1924 | 2309 | 2694 | 386 

053078 | 3463 | 3846 | 4230 | 4613 || 4996 | 5878 | 5760 | 6142 | 6524 | 383 

6905 | 7286 | 7666 | 8046 | 8426 || 8805 | 9185 | 9563 | 9942 |——— : 
0320 i 

1075 | 1452 | 1829 | 2206 || 2582 | 2958 | 3333 | 3709 | 4088 | 376 
4458 | 4832 | 5206 | 5580 | 5953 || 6326 | 6699 | 7071 | 7443 | 7815 | 38% 

8186 | 8557 | 8928 | 9298 | 9668 



























































0038 | 0407 | O776 | 1145 | 1514 | 37 
071882 | 2250 | 2617 | 2985 | 8352 || 3718 | 4085 | 4451 | 4816 | 5182 | 366 
5547 | 5912 | 6276 | 6640 | 7004 || 7368 | 7731 | 8094 | 8457 | 8819 | 363 

















om soo Pw we 
° 
& 
> 
S 
o 
(ea) 











PROPORTIONAL PARTS. 




















Diff 1 2 3 4 5 6 7 8 9 
895 | 39.5] 79.0 118.5 | 158.0 | 197.5 | 287.0 | 276.5 | 316.0 | 355.5 
894 | 39.4 | 78.8 118.2 | 157.6 | 197.0 | 236.4] 275.8] 3815.2 | 354.6 
393} 39.3 | 78.6 D170 15732 a 196.071 230.5 "5.1 | 3814.4 | 353.7 
392 | 39.2 | 78.4 117.6 | 156.8 | 196.0 | 2385.2 74.4 | 313.6°| 352.8 
Ole Poo ale| teres 117.3 |. 156.4 | 195.5 | 284.6") 273.7 | 3812.8 | 351.9 
390 | 89.0 | %8.0 117.0 | 156.0) 195.0 | 234.0 | 278.0 | 312.0 | 351.0 
Boge) Oso Mie 116.7 | 155.6 | 194.5 | 283.4 72.8 | 811.2 | 350.1 
388 | 388.8 | 77.6 116.4 | 155.2) 194.0 | 282.8] 271.6] 310.4 | 349.2 
387 | 88.7 | 77.4 116.1 | 154.8] 193:5 | 282.2) 270.9 | 309.6 | 348.3 
3886 | 38.6 | 77.2 115.8 | 154.4] 198.0 | 231.6 | 270.2 | 3808.8 | 347.4 
3885 | 38.5 | V7.0 115.5 | 154.0 | 192.5 | 231.0} 269.5 | 308.0 | 346.5 
384 | 38.4 | 76.8 115.2 | 153.6 | 192.0 | 230.4 | 268.8 | 307.2 | 345.6 
383 | 88.3 | 76.6 114.9 | 153.2 | 191.5 | 229.8 | 268.1 | 306.4 | 344.7 
382 | 38.2 | 76.4 114.6 | 152.8 | 191.0 | 229.2} 267.4 | 305.6 | 343.8 
3881 | 388.1] 76.2 114.3 |° 152.4 |: 190.5 | 228.6 | 266.7 | 304.8 | 342.9 
380 | 38.0} 76.0 114.0 | 152.0] 190.0 | 228.0} 266.0 | 304.0 | 342.0 
379 | 87.91 75.8 118.7 |° 151.6 | 189:5 | 227.4 | 265.81 303.2 | 341.1 
37 57.8 | 75.6 113.4 | 151.2 | 189.0 | 226.8 | 264.6 | 302.4 | 340.2 
37 30.7 | 75.4 118.1 | 150.8 | 188.5 | 226.2 | 268.9 | 301.6 | 839.3 
Ome oucOs) Cane 112.8 | 150.4 | 188.0 | 225.6 | 2638.2 | 300.8 | 3388.4 
(On | OC 10.0 112.5 | 150.0 | 187.5 | 225.0 | 262.5 | 300.0 , 3387.5 
374 | 387.4 | 74.8 112.2 | 149.6 | 187.0 | 224.4 |] 261.8 | 299.2 | 336.6 
3873 | 87.3 | 74.6 111.9 | 149.2 | 186.5 | 228.8 | 261.1 | 298.4 | 335.7 
3% 87.2 | 74.4 111.6 | 148.8 | 186.0 | 223.2 | 260.4 | 297.6 | 334.8 
3871 | 87.1 | 74.2 111.8 | 148.4 | 185.5 | 222.6 | 259.7 | 296.8 | 3338.9 
370 | 387.0 | 74.0 111.0 | 148.0} 185.0 | 222.0 | 259.0 | 296.0 | 3383.0 
369 | 86.9} 73.8 110.7 | 147.6 | 184.5 | 221.4 | 258.3 | 295.2 | 382.1 
368 | 386.8] 73.6 110.4 | 147.2 | 184.0 | 220.8 | 257.6 | 294.4 | 331.2 
367 | 386.7 | 73.4 110.1 | 146.8 | 183.5 | 220.2 | 256.9 | 293.6 | 330.3 
366 | 86.6 | 73.2 109.8 | 146.4 | 183.0 | 219.6 | 256.2 | 292.8 | 320.4 
365 | 86.5] 73.0 109.5 | 146.0 | 182.5 | 219.0 | 255.7 | 292.0 | 328.5 
364 | 36.4] 72.8 109.2 | 145.6 | 182.0 | 218.4 | 254.8 | 291.2 | 327.6 
363 | 36.3 | 72.6 108.9 | 145.2] 181.5} 217.8 | 254.1 | 290.4 | 326.7 
362 | 36.2 | 72.4 108.6 | 144.8] 181.0] 217.2] 253.4 | 289.6 | 325.8 
361 | 36.1 | 72.2 108.3 | 144.4] 180.5] 216.6 | 252.7 | 288.8 | 824.9 
360 | 36.0 | 72.0 108.0 | 144.0 180.0] 216.0 | 252.0 | 288.0 | 324.0 
359 | 85.9] 71.8 107.7 | 143.6 79.5 | 215.4 | 251.38 | 287.2 | 823.1 
358 | 35.8] 71.6 107.4 | 148.2 | 179.0] 214.8 | 250.6 | 286.4 | 322.2 
857. | 385.7 | 71.4 107.1 | 142.8 | 178.5] 214.2 | 249.9 | 285.6 | 321.3 
856 | 35.6 | 71.2 106.8 | 112.4 | 178.0 | 218.6} 249.2 | 284.8 | 820.4 











TABLE XXIV.—LOGARITHMS OF NUMBERS 





" No. 120 L. 079. 


N, 





r= 
*0o 
RWW HOO MIO TP wde-e 
















































































65. 


[No. 134 L. 180. 
5 6 7 8 9 






















































































0 if 2 4 Diff. 
079181 | 9543 | 9904 }—|_—— | |, ——— | | ——_ | ——__ | —_|__ 
—— |——- | 0266 | 0626 || 0987 | 1347 | 1707 | 2067 | 2426 360 
082785 | 3144 | 3503 | 3861 | 4219 || 4576 | 4934 | 5291 | 5647 | 6004 857 
ae 6716 | 7071 | 7426 | 7781 |} 8186 | 8490 | 8845 | 9198 | 9552 855 
———_| 0258 | 0611 | 0963 | 1315 || 1667 | 2018 | 2370 | 2721 | 3071 352 
093422 | 3772 | 4122 | 4471 | 4820 |! 5169 | 5518 | 5866 | 6215 | 6562 349 
6910 | 7257 | 7604 | 7951 | 8298 || 8644 | 8990 | 9335 | 9681 | as 
Sete ———- ee 0026 346 
100371 | 0715 | 1059 | 1403 | 1747 || 2091 | 2434 | 2777 | 3119 | 3462 343 
3804 | 4146 | 4487 | 4828 | 5169 ||.5510 | 5851 | 6191 | 6531 | 6871 341 
7210 | 7549 ' 7888 | 8227 | 8565 || 8903 | 9241 | 9579 | 9916 |— 
——s ——_—_}—__—— - 0253 838 
110590 | 0926 | 1263 | 1599 | 1934 || 2270 | 2605 | 2940 | 3275 | 3609 335 
3943 | 4277 | 4611 | 4944 | 5278 || 5611 | 5943 | 6276 | 6608 | 6940 goo! 
7271 | 7603 | 7984 | 8265 | 8595 || 8926 | 9256 | 9586 | 9915 
= 0245 330 
120574 | 0903 |. 1231 | 1560 | 1888 || 2216 | 2544 | 2871 | 3198 | 3525 3828 
3852 |.4178 | 4504 | 4830 | 5156 5451 | 5806 | 61381 | 6456 | 6781 3825 
7105 | 7429 | 7753 | 8076 | 8399 || 8722 | 9045 | 9368 | 9690 
13 0012 323 
PROPORTIONAL PARTS. 
1 2 3 4 5 6 v6 8 9 
35 bee CLO 106.5 | 142.0] 177.5 | 213.0 | 248.5] 284.0 | 319.5 
35.4 |. 70.8 106.2 | 141.6 | 177.0 | 212.4) 247.8 | 283.2 | 318.6 
35 a0) 10_6 WOjsGeeeltlean| al ;GcOnetelieou sett las ose aah ol yey 
Sezai le: 105.6 | 140.8} 176.0 | 211.2 | 246.4 | 281.6 | 316.8 
35.1 | 70.2 105.3 | .140.4 175.5 } 210.6 | 245.7 | 280.8 | 315.9 
35.0! 70.0 105.0 | 140.0.) 175.0! 210.0 | 245.0] 280.0 ! 315.0 
34.9 | 69.8 104-7°| 189.67) -174.5.| 209.4 | 244.3) 279.2 | 8144 
34.8 | 69.6 104.4} 139.2; 174.0) 208.8 |: 248.6} 278.4 | 313.2 
34.7! 69.4 ‘104.1 138.8 | 173.5 | 208.2 | 242.9] 277.6 | 312.3 
34.6 | 69.2 OSES 2) cel eseta| - 173208) 120768) | 242.281 27658) Bites 
34.5 | 69.0 108.5.) 188,07) 17255.) 207-0-| 241.5 | 276.0 | 310.5 
34.4 | 68.8 193:2 | 187-6 | 172.0 | 206.4) 240.81 275.2 | 309.6 
84.3.) 68.6 102.9 | 137.2 (1:5 | 205.81 240.1 |, 27424 | 30827 
84025) 68.4 102.6} 186.8 | 171.0 | 205.2) 239.4] 278.6 | 307.8 
34.1 | 68.2 102:3|- 186.4 | 170.5 | 204.6 | 288.7] 272.8 | 306.9 
34.0 | -68.0 10250) 186-9.) 170.0)! 20450 | 238-0:)| 27270. | 306.0 
33.9 | 67.8 LOM Cs ea sonGal 6169. Oe! | 0se4el Senge ge) meres Aiped 
3826 7be 67.6 100-4 | 185.24) 169.0") 202.8 286-6 | 270.4 | 304-2 
32172). 67.4 101.1 134.8 | 168.5 | 202.2 | 285.9 | 269.6 | 303.3 
SozGal.. Otee 100.8 | 184.4] 168.0 | 201.6 | 285.2] 268.8 | 302.4 
So ate OLeG 100.5 | 134.0] 167.5 | 201.0 | 2384.5] 268.0] 301.5 
33.4 | 66.8 100.2} 133.6] 167.0 | 200.4] 283.8 | 267.2 | 300.6 
S5.501) BOLO 9979") 183.2) 166:5') 199.8 | 288.1) 266.4 | 299.7 
SEPP a ea ofok! 99.6; 182.8 | 166.0 | 199.2] 282.4 | 265.6 | 298.8 
33.1} 66.2 99.3 |° 182.4 | 165.5 | 198.6] 231.7) 264.8 | 297.9 
33.0 | 66.0 99.0 | 182.0! 165.0 |. 198:0! 281.0! 264.0 | 297.0 
382.9 | 65.8 98.7 | 181.6 | -164.5 | 197.4} 280:3 | 263.2 |) 296.1 
32.8 | 65.6 - 98.4 | 181.2] 164.0} 196.8 | 229.6 | 262.4 | 295.2 
Seca 65.4 98.1 130.8 | 168.5 | 196.2] 228.9] 261.6 | 294.3 
82.6 | 65.2 97.8 | 180.4 | 163.0] 195.6} 228.2 | 260.8 | 293.4 
$2.5_|, 65.0 97.5 | 180.0} 162.5 | 195.0] 227.5] 260.0 | 292.5 
32.4 64.8 97.2 129.6 162.0 194.4 226.8 959.2 | 291.6 
82.3 | 64.6 96.9 129.2 161.5 193.8 226.1 258.4 | 290.7 
82.2 | 64.4 96.6 | 128.8 | 161.0 | 198.2 | 225.4 | 257.6 | 289.8 





66 


TABLE XXIV.—LOGARITHMS OF NUMBERS 





No. 135 L. 130.] 


[No, 149 L. 1%, 






























































































































































N. 0 1 2 3 4 5 6 8 9 | Diff. 
135 | 130334 | 0655 | 0977 | 1298.] 1619 || 1939 | 2260 | 2580 | 2900 | 8219 321 
6 3539 | 8858 | 4177 | 4496 | 4814 || 5133 | 5451 | 5769 | 6086 | 6403 318 
v 6721 | 7037 | 7854 | 7671 | 7987 || 8803 | 8618 | 8934 | 9249 | 9564 316 
8} 9879 |—— oa - 
————]| 0194 | 0508 | 0822 | 1136 || 1450 | 1763 | 2 2389 | 2702 314 
9 | 143015 | 33827 | 3639 | 3951 | 4268 || 4574 | 4885 | 5196 | 5507 | 5818 311 
140 6128 | 6438 | 6748 | 7058 | 7367 || 7676 | 7985 | 8294 | 8603 | 8911 809 
al 9219 | 9527 | 9885 — —— 
———}-__—|.___|. 0143 | 0449 || 0756 | 1063 | 1370 | 1676°| 1982 807 
2 | 152288 | 2594 | 2900 | 3205 | 3510 || 8815 | 4120 | 4424 | 4728 | 5082 805 
8 | 5336 | 5640 | 5948 | 6246 | 6549 || 6852 | 7154 | 7457 | 7759 | 8061 303 
4 8362 | 8664 | 8965 | 9266 | 9567 || 9868 —|——— ee 
Senet ae eee —|/——| 0168 | 0469 769 | 1068 301 
5 | 1613868 | 1667 |- 1967 | 2266 | 2564 || 2863 | 3161 | 3460 | 38758 | 4055 299 
6 4353 | 4650 | 4947 | 5244 | 5541 || 58388 | 6134 | 6480 | 6726 | 7022 297 
v 7317 | 7613 | 7908 | 8203 | 8497 || 8792 | 9086 | 9380 | 9674 | 9968 295 
8 70262 | 0555 | 0848 | 1141 | 1434 1726 | 2019 | 2311 | 2603 | 2895 293 
9 3186 | 3478 | 8769 | 4060 | 4351 || 4641 | 4982 | 5222 | 5512 | 5802 291 
PROPORTIONAL PARTS. 
Diff. 1 2 3 4 5 6 7 8 9 
821 32.1 64.2 96.3 128.4 160.5 192.6 224.7 256.8 | 288.9 
$201 32-0) | 64:0 96.0 128.0 | 160.0] 192.0 | 224.0 | 256.0 | 288.0 
819 | 81.9! 63.8 95.7 12756.) 1h9soul) HOLA S83 851" O55 02 sean el 
318 | 8t.8 | 63.6 95.4 12752). 159707) 190:85F 222"65) ~2h4 4 |) 2kbr2 
317 Oley 63.4 95.1 126.8 158.5 190.2 221:9 253.6 je coons 
DIG ol Gn Oars 94.8 126.4.) 158.0] 189.6.) 221.2! 252.8 | 284.4 
815 | 31:5). 63:0 94.5 126.0; 157.5 | 189.0} 220.5 | 252.0 | 283.5 
314 | 31.4] 62.8 94.2 125.6 | 157.0} 188.4) 219.8 | 251.2 | 282.6 
Stage ol a oecO 93.9 125 2) 156255) ASH8 i 219 Se e250 aa oot 
Blew ivoleor is 02.4. 93.6 124.8 | 156.0) 187.2 | 218.4 | 249.6] 280.8 
Stile |) slits) 6272 93.3 124.4 | 155.5] 186.6] 217.7 | 248.8 | 279.9 
310 | 31.0] 62.0 93.0 124:0 | ' 155:0.| 186.0] 217.0] 248:0 | 279.0 
3809 | 30.9] 61.8 92.7 12336.) 154/534) 185240) 216-35) 247 o0) oi eal 
3808 | 30.8 | 61.6 92.4 123752 1 . 1540 5|- 184731) 215 Gale 246.4) aie 
Beier OO tall) Ole. 92.1 192784) 158201" 184 90 Ota or) 245.67) enoes 
306° | 380.6.| - 61.2 91.8 1924 15-0 = LS3.00) | Blea t ore 
305 | 30.5 | 61.0 91.5 122.0 |- 152.5} 183.70 |. 213.541 24470 | 2745 
304 | 30.4 | 60.8 91.2 131 Go) 152-0g) O24 ele Olesen eas 
303 30.3 60.6 90.9 IPAS 15105 181.8 212.1 242.4 | Sion 
802 | 80.2 | 60.4 90.6 12058 | 151-0 ISP Sc) B14) Oat Pees 
801 | 30:1 | 60.2 90.3 120.4 | 150.5) 180.6 | 210.7 | 240.8) 270.9 
800 | 30.0 | 60.0 90.0 120.0 | 150.0}. 180.0 | 210.0} 240.0 | 270.0 
299 | 29.9 | 59.8 89.7 119.6 | 149.5 | 179.4} 209.3 | 289.2 | 269.1 
298 | 29.8] 59.6 89.4 119.2 | 149.0 | 178.8 | 208.6 | 288.4 | 268.2 
297 | 29.7 | 59.4 89.1 1188) 148°5 1) 178.2 207.9.) 237.6 1 26725 
296 | 29.6 | 59.2 88.8 118.4 |} 148.0 | 177.6 | 207.2 236.8 | 266.4 
295 | 29.5 | 59.0 88.5 118.0°} 147.5 | .177.0:| 206.5 | 2386.0 | 265.5 
294 | 2974) 5a58 88.2 117-6) 1470) 176.4) 20a28 a) 235.2 264.6 
293 | 29.3) 58.6 87.9 117.2) 146.5 | 175.8} 205.1.) 234-4 | 263.7 
292 | 29.2] 58.4 87.6 116.8 | 146.0} 175.2 | 204.4) 283.6 | 262.8 
291 | 29.1] 58.2 87.3 116.4 | 145.5 174.6) 203.7) 232.8 | 261:9 
290 | 29.0] 58.0 87.0 116.0 | 145.0] 174.0) 2038.0} 282.0 | 261.0 
289 | 28.9] 57.8 86.7 115.6 | 144.5.) 173.41) 2023.) 231.21 260.4 
288 | 28.8 |, 57.6 86.4 115.2 | 144.009, 172787) )20ie Gal) 23024, 7-25 9s2 
Ove et Paseo alsitiae § 86.1 114:8 | 143.5), 172.2 | “200,9)) 229.6 25823 
286 "1, 28.6.1 57.2 85.8 114 Ay): 143.04 Ta Ga e200. 2al- 228-8.) cages 


TABLE XXIV.—LOGARITHMS OF NUMBERS 





67 
















































































































































































No. 150 L. 176.] [No. 169 L. 230, 
N. 0 1 2 3 4 65 a 8 9 | Diff. 
150 | 176091 | 6381 | 6670 | 6959 | 7248 || 7586 | 7825 | 8113 | 8401 | 8689 | 289 
1 8977 | 9264 | 9552 | 9839 |— = eee 
pee ee: 0126 || 0418 | 0699 | 0986 | 1272 | 1558 | 287 
2 | 181844 | 2129 | 2415 | 2700 | 2985 || 8270 | 38555 | 3839 | 4123 | 4407 | 285 
3 4691 | 4975 | 5259 | 5542 | 5825 || 6108 | 6391 | 6674 | 6956 | 7239 | 283 
4 7521 | 7803 | 8084 | 83866 |! 8647 8928 | 9209 | 9490 | 97°71 
a = —.- ——_ 0051 | 281 
5 | 190332 | 0612 | 0892 | 1171 | 1451 730 | 2010 | 2289 | 2567 | 2846 | 279 
6 3125 | 3403 | 8681 | 3959 | 4237 || 4514 | 4792 | 5069 | 5346 | 5623 278 
vi 5900 | 6176 | 6453 | 6729 | 7005 || 7281 | 7556 | 7882 | 8107 | 8882 | 276 
8 8657 | 8932 | 9206 | 9481 | 9755 5 
eS ee 029) 10803) 0577 | (0850! |) 11241 a74 
9 | 201897 | 1670 | 1943 | 2216 | 2488 || 2761 | 3088 | 38305 | 3577 | 3848 | 272 
160 4120 | 4891 | 4663 | 4984 | 5204 || 5475 | 5746 | 6016 | 6286 | 6556 | 271 
1 6826 | 7096 | 73865 | 7684 | 7904 || 8173 | 8441 | 8710 | 8979 | 9247 | 269 
“2 9515 | 9783 
—|————| 0051 | 0319 | 0586 || 0853 | 1121 | 1888 | 1654 | 1921 | 267 
8 | 212188 | 2454 | 2720 | 2986 |: 8252 3518 | 3788 | 4049 | 4814 | 4579 266 
4 4844 | 5109 | 5873 | 56388 | 5902 || 6166 | 6480 | 6694 | 6957 | 7221 | 264 
5 7484 | 7747 | 8010 | 8273 | 8536 798 | 9060 | 9823 | 9585 | 9846 | 262 
6 | 220108 | 0370 | 0631 | 0892 | 1153 || 1414 1936 | 2196 | 2456) 261 
4 2716 | 2976 | 8286 | 3496 | 3755 || 4015 | 4274 | 4583 | 4792 | 5051 | 259 
8 5309 | 5568. | 5826 | 6084 | 6342 || 6600 | 6858 | 7115 | 7872 | 7680 | 258 
9). 7887 | 8144 | 8400 | 8657 | 8913 || 9170 | 9426 | 9682 | 99388 |—— 
23 | 0198 | 256 
PROPORTIONAL PARTS. 
Diff. 1 2 3 4 5 6 i 8 9 
2285 | 28.5 7.0 85.5 114.0 | 142.5 1.0} 199.5 | 228.0 | 256.5 
284 28.4 56.8 85.2 11336 142.0 0.4 198.8 2272 | 255.6 
283. | 28.3 | 56.6 84.9 113.25) 145 9.8 | 198.1 226.4 | 254.7 
2982 |°28.2 | 56.4 84.6 lp jated Ml ak Gha0) 9.2'| 197%.4) 22526: | 253.8 
281 | 28.1 56.2 84.3 1124 | 140.5 | 168.6 196.7 | 224.8 | 252.9 
280 | 28.0) 56.0 84.0 112.0 | 140.0! 168.0] 196.0 | 224.0 | 252.0 
279 | 27.9 | 55.8 83.7 sna), sile\)atsy ala taheyears il) winkeye) | eo 2eeaiaee leas Wal 
278 27.8 55.6 83.4 111.2 139.0 166.8 194.6 rdpeaeal I Alsi) 
ie wletiet ut tio. 4 83.1 110.8 13825. |) 166.2" |) 193809 |. 22T6i 24973 
270 1n27.60)| Dee 82.8 TIO el alos.Or | pelOROu ln Ose) |e eeO On meds: 
05 wes |, 5520 82.5 110.0 | 187.5) 165.0 | 192.5 | 220.0 | 247.5 
O14 (02%e4 |. B4i8 82.2 109.6 | 187.0) 164.4 | 191.8 | 219.2 | 246.6 
Sia tle eiaoal) 64.6 81.9 109.2 SOM ae A Looes al Lo Lele y welt mets | 
O72) Toyo 54.4 81.6 108.8 | 186.0 | 168.2) 190.4 | 217.6 | 244.8 
Cilla ehetaly Olas 81.3 108.4 135.5 | 162.6 |) 189:7 | 216°8 | 243.9 
QF 27.0 |. 54.0 81.0 108.0 | 135.0 | 162.0 | 189.0} 216.0 | 243.0 
269 | 26.9] 53.8 80.7 107.6 eA Oler i iadeGronlt Letoeoml mere 
268 | 26.8 | 53.6 80.4 LOZe20| dot cOnls BOSE TAS 6 eae 2 
Ore Wl eOng tease 80.1 106.8 | 1838.5 | 160.2; 186.9 | 213.6 | 240.3 
PROBE eebe Or) moose 79.8 ieee OE VES) t0) 159.6 | 186.2) 212.8 | 239.4 
265 | 26:5.) 53.0 79.5 106.0 | 182.5] 159.0} 185.51. 212.0 | 288.5 
264 | 26.4] 52.8 79.2 105.6 | 182.0 |: 158.4] 184.8) 211.2 | 237.6 
O66 wlaebdo |) oes 78.9 105.2-| 181.5 | 157.8 | 184.1 |) 210.4) 23627 
Posen eeOsenl poe 78.6 104.8 | 1381.0) 157.2 | 183.4 | 209.6 | 285.8 
261 26.1 52.2 ws 78.3 104.4 130.5 156.6 182.7 208.8 | 234.9 
260 | 26.0] 52.0 78.0 104.0 | 130.0 | 156.0] 182.0 | 208.0 | 234.0 
259 91 °25.01 1° 5158 el 103261) 112075 | 155.4 |) 1181.37) 80752) | 2838.1 
258 | 25.8 | 51.6 77.4 103.2 | 129.0) 154.8 | 180.6 | 206.4 | 282.2 
Ol EoD a Weal sce ion TO2GS Te e8ed |) 15462" |) ZOO) 1205.6) | e2sieo 
256 1°25,6. | 5122 76.8 102.4 128.0 | 153.6 | 179.2 | 204.8 | 230.4 
DoD Mol tpl sO 7650 102.0.| 127.5 | 153.0] 178.5 | 204.0} 229°5 





TABLE XXIV.—LOGARITHMS OF NUMBERS 






































1 





0704 
8250 
5781 
8297 


0799 
3286 
5759 
8219 


0664 
3096 
5514 
7918 


0310 
2688 
5054 
7406 
9746 


2074 
4389 
6692 











2 


SDD BODOODSOSSSCSrH 


SASSSSSES SSSRRarAIAE 


IIR ‘ 
WARAWOWRADWD CORMDWOWRADWO 


ee 
J 


68 

No. 170 L. 230.] 
N.| 0 
170 | 230449 
1| 2996 
2] 5528 
3] 8046 
4 | 240549 
5| 3038 
6| 5513 
7) 7973 
8 | 250420 
9 | 2853 
180 | 527 
1) 7679 
2 | 260071 
3] 2451 
4| 4818 
5 | vie 
6| 9513 
7 | 271842 
8} 4158 
9 | 6462 
Diff.| 1 
255 | 25.5 
254 | 25.4 
253 | 25.3 
252 | 25.2 
251 | 25.1 
250 | 25 0 
249 | 24.9 
248 | 24.8 
B47 | 24.7 
246 | 24.6 
R45 | 24.5 
244 | 24.4 
243 | 24.3 
242 | 24.2 
241 | 24.1 
240 | 24.0 
239 | 23.9 
238 | 23.8 
237 | 23.7 
236 | 23.6 
235 | 23.5 
234 | 23.4 
233 | 23.3 
232 | 23.2 
231 | 23.1 
230 | 23.0 
229 | 22.9 
228 | 22.8 
227 | 22.7 
226 | 22.6 






































































































































[No. 189 L. 278. 
2 3 4 6 6 vi 8 | 9 | Diff. 
0960 | 1215 | 147 1724 | 1979 | 2234 | 2488 | 2742 255 
8504 | 3757 |. 4011 4264 | 4517 | 477 5023 | 5276 253 
6033 | 6285 | 6537 || 6789 | 7041 | 7292 | 7544 | 7795 | 252 
8548 | 8799 | 9049 || 9299 | 9550 | 9800 
es — 0050 | 0300 } 250 
1048 | 1297 | 1546 || 1795 | 2044 | 2293 | 2541 | 2790 | 249 
3534 | 3782 | 4030 || 4277 | 4525 | 4772 | 5019 | 5266 | 248 
6006 | 6252 | 6499 || 6745 | 6991 | 7237 | 7482 | 7728 | . 246 
8464 | 8709 | 8954 |} 9198 | 9443 | 9687 | 9932 }——— 
a Se 0176 | 245 
0908 | 1151 | 1395 || 1638 | 1881 | 2125 | 2368 | 2610 | 243 
38338 | 3580 | 3822 || 4064 | 4306 | 4548 | 4790 | 5081 | 242 
750 | 5996 | 6237 || 6477 | 6718 | 6958 | 7198 | 7439 | 241 
8158 | 8398 | 8637 || 8877 | 9116 | 9355 | 9594 | 9833 | 239 
0548 | 0787 | 1025 || 1263°| 1501 | 1739 | 1976 | 2214 | 238 
2925 | 3162 | 3399 || 3636 | 3873 | 4109 | 4346 | 4582 | 237 
5290 | 5525 | 5761 || 5996 | 6232 | 6467 | 6702 | 6937 | 235 
heh "875 | 8110 ||} 8344 | 8578 | 8812 | 9046 | 927 234 
9980 — _|—__}—+___ —- 
0213 | 0446 || 0679 | 0912 | 1144 | 1377 | 1609 | 233 
2306 | 25388 | 277 8001 | 3233 | 3464 | 3696 | 3927 | 232 
4620 | 4850 | 5081 5311 | 5542 | 5772 | 6002 | 6232 230 
6921 | 7151 | 7380 7609 | 7838 | 8067 | 8296 | 8525 229 
PROPORTIONAL PARTS, 

“34 4 5 6 ff 8 9 
76.5 102.0} 127.5] 158.0] 178.5 | 204.0 | 229.5 
76.2 101.6 27-0 1 15234 I) 17728 | > 208.2 22826 
vitor) 101.2 126.5 151.8 177.1 202.4 | 227.7 
75.6 10028 | p42670 1 2 15122 | pa7674 |) 2OU6s=226.8 
95.3 100.4 25.5 150.6") 175.27 | | -200.851-225 49 
"5.0 100-0 |.) 125.0 | 150.0.) 175.0) > 200208) 225.0 
iol 99.6] 124.5 | 149.4 |) 17473 1. 19952)) oan Ft 
%4.4 9922-1240 15 148.8 4) 5.6 | osetelscenae 
"4.1 98-8" | 2123. 5 = 14828 | (ge aie O baleen o 
"3.8 98.54 °|_ 12370 1. 14766") 19222 || 2 sf9658 224. 
"305 98:0) 122.5 }. 14770") + IgA 25 _) > 29650"7 22055 
ane ".6 | 122.0] 146.4} 170.8 | 195.2 | 219.6 
72.9 ee EEN tes} 145.8 170.1 194.4 | 218.7 
72.6 96.8 | 121.0 | 145.2 | 169.4; 1938.6 | 217.8 
Tepe 96.4 | 120.5] 144.6] 168.7 | 192.8 |) 216.9 
72.0 96.0 | 120.0] 144.0} 168.0] 192.0 | 216.0 
Lg Weer 9556 I 119.5 |) 1435471 § 16708. 0 191 aie ta 
AGA 95.2) 119.0] 142.8 | 166.6 | 190.4} 214.2 
1.1 04°8 | 118.5 | 14272) 165.9), 189°6) 21323 
70.8 94.4 | 118.0} 141.6 | 165.2] 188.8 | 219.4 
"0.5 9450} 117.5 | 141.0 | 16455 | 28850") 211-5 
70.2 93:6) 117.0} 140.4]. 163.8 | 187-2 | 210.6 
69.9 938.2 | 116.5 | 189.8 | 1638.1 186.4 | 209.7 
69.6 92.8 | 116.0 |}. 1389.2} 162.4] 185.6 | 208.8 
69.3 92:4 | 115.5 | 138.6.) ) 161.7% | 184-8) 207.9 
69.0 92.0 | 115.0] 188.0} 161.0] 184.0 | 207.0 
68.7 OF36.| -11455 | 13%cdo es 60 tsp 163852.) 20621 
68.4 91.2 | 114.0 | 186.8 | 159.6 | 182.4 | 205.2 
68.1 90.8 | 118.5 | 186.2 |-158.9 | 181.6 | 204.8 
67.8 90:4 | 113.0} 1385.6 | 158.2 } 180.8 | 208.4 


TABLE XXIV.—LOGARITHMS OF NUMBERS 


69 





No. 190 L. 278] 


























n| 0 
ee 
190 
1 | 281033 
21° 3301 
3 5557 
4 7802 
5 | 290085 
6 2256 
7 4466 
&: 6665 
0 | 8853 
200 | 301080 
= etl 3196 
2 53864 
3. 7496 
4| 9630 
5 | 311754 
6 38867 
7 | 5970 
“g | 8063 
9 | 320146 
O10 2219 
fq 4282 
2] 6336 
3 | 8380 
4 | 330414 
pitt.| 1 | 
995 99 5 
994 99 4 
223 ppaes) 
222 pe) 
921 Ae | 
220 22.0 
219 21.9 
218 21.8 
O17 DIGG 
216 21.6 
215 Sin 
214 21.4 
213 21.3 
212 a 
911 Ot a 
210 21.0 
209 | 20.9 
208 | 20.8 
207 20.7 
206 | 20.6 
205 20.5 
204 | 20.4 
203 | 20.3 
20.2 











0617 





SSSEHEEEE SRSRSSSS SSESEELS 
PADOWRRD OCVURAWDOWR AWDOWRADWS 





1464 | 1681 
3628 | 3844 
5781 | 5996 
7924 | 8137 














0819 | 1022 


























ooo 














1427 

















1630 









































[No. 214 L, 382. 
731 8 
0351 | 0578 
2622 | 2849 
4882 | 5107 
7130 | 7354 
9366 | 9589 
1591 | 1813 
3804 | 4025 
6007 | 6226 
8198 | 8416 
0378 | 0595 
2547 | 2764 
4706 | 4921 
6854 | 7068 
8991 | 9204 
1118 | 1330 
8234 | 3445 
53840 | 5551 
7436 | 7646 
9522 | 9730 
1598 | 1805 | 2012 | 207 
3665 | 3871 | 4077 | 206 
5721 | 5926 | 6131 205 
767 | 7972 | 8176 | 204 
9805 |-—— 
——J 0008 | 0211 | 203 
1832 | 2034 | 2236 | 202 
8 9 
.0| 157.5} 180.0 | 202.5 
A 156.8 179.2 | 201.6 
'8| 156.1 | 178.4 | 200.7 
‘2 | 155.4] 177.6 | 199.8 
6 154 176.8 | 198.9 
0 | 154.0] 176.0 | 198.0 
4} 153.3| 175.2 | 197.1 
‘8 | 152.6] 174.4 | 196.2 
.2| 151.9 | 173.6 | 195.3 
6 151.2 172.8 | 194.4 
0 ‘150.5 172.0 | 1938.5 
4 149.8 171.2 | 192.6 
8 149.1 170.4 | 191.7 
‘2 | 148.4 |- 169.6 | 190.8 
‘6 | 147.7. | 168.8 | 189.9 
0 | 147.0} 168.0 | 189.0 
4} 146.3| 167.2 | 188.1 
8| 145.6 | 1664 | 187.2 
'2| 144.9 | 165.6 | 186.3 
6| 144.2 | 164.8 | 185.4 
0 | 143.5 | 164.0 | 184.5 
-4| 142.8 | 163.2 | 183.6 
‘8 | 142.1 | 162.4 | 182.7 
2] 141.4] 161.6 | 181.8 








70 


No. 215 L. 332.] 


4 


TABLE .XXIV.—LOGARITHMS OF NUMBERS 





























N. 0 
215 | 332438 
6 4454 
7 6460 
8 8456 
9 | 340444 
220 2423 
1 4392 
2 6353 
3 8305 
4 | 350248 
5 2183 
6 4108 
7 6026 
8 7935 
9 9835 
230 | 361728 
1 3612 
R 5488 
3 7356 
4 9216 
5 | 871068 
6 2912 
7 4748 
8 6577 
9 8398 

38 
. 

Diff 1 
202 }.20.2 
201 } 20.1 
200 | 20.0 
199 | 19.9 
198 | 19.8 
197; | 19.7 
196 | 19.6 
-195 | 19.5 
194 | 19.4 
193 | 19.3 
192 | 19.2 
191 | 19.1 
190 | 19.0 
189 | 18.9 
188 , 18.8 
187 | 18.7 
186 | 18.6 
185 | 18.5 
184 | 18.4 
183 | 18.3 
182 | 18.2 
18ti)) 18.4 
180 "| 18.0 
179 | 17.9 





















































[No. 239 L. 380, 



































2 3 4 5 6 7 | 8 | 9 | Diff. 
2842 | 3044 | 3246 || 3447 | 8649 | 3850 | 4051 | 4253 202 
4856 | 5057 | 5257 || 5458 | 5658 | 5859 | 6059 | 6260 201 
6860 | 7060 | %260 || 7459 | 7659 | 7858 | 8058 | 8257 200 
8855 | 9054 | 9253 || 9451 | 9650 | 9849 | ——— 
Ss —| |——-— | —_~—-|——| 0047 | 0246 199 
0841 | 1039 | 1237 || 1485 | 1682 | 1880 | 2028 | 2225 | 198 
2817 | 8014 | 3212 || 8409 | 38606 | 3802 | 3999 | 4196 | 197 
785 | 4981 | 5178 || 5374 | 557 5766 | 5962 | 6157 196 
6744 | 6989 | 7135 || 73830 | 7525.) 7720 | 7915 | 8110 195 
8694 | 8889 | 9083 || 9278 | 9472 | 9666 | 9860 | —— 
ee —|—-—|_—_— |__| 0054 | 194 
0636 | 0829 | 1023 1216 | 1410 | 1603 796 | 1989 193 
2568 V61 | 2954 || 8147 | 3339 | 8532 | 8724 | 3916 193 
4493 | 4685 | 4876 || 5068 | 5260 | 5452 | 5643 | 5884 192 
6408 | 6599 | 6790 || 6981 | 7172 | 7363 | 7554 | 7744 | 191 
8816 | 8506 | 8696 || 8886 | 9076 | 9266 | 9456 | 9646 190 
0215 | 0404 | 0593 || 0783 | 0972 | 1161 | 1850 | 1589 | 189 
2105 | 2294 | 2482 || 2671 | 2859 | 3048 | 8286 | 3424 | 188 
38988 1 4176 | 4863 || 4551 | 4789 | 4926 | 5113 | 53801 188 
5862 | 6049 | 6236 || 6423 | 6610.4 6796 | 6988 | 7169 187 
7729 | 7915 | 8101 8287 | 8473 | 8659 | 8845 | 9030 186 
9587 | 9772 | 9958 = =) rie a Sa 
——— | ——_— |————__|| 0143 | 0828 | 0513 | 0698 | 08838 185 
1437 | 1622 | 1806 1991 | 2175 | 2860 | 2544 | 2728 184 
8280 | 3464 | 3647 || 8831 | 4015 | 4198 | 4882 | 4565 184 
5115 | 5298 | 5481 5664 | 5846 | 6029 | 6212 | 6394 183 
.6942 | 7124 | 7306 || 7488 | 7670 | 7852 | 8084 | 8216 182 
8761 | 8948 | 9124 || 9306 | 9487 | 9668 | 9849 |—_— 
0030 181 
PROPORTIONAL PARTS. 
3 4 5 6 % 8 9 
60.6 80.8 101.0 122 141.4 161.6 | 181.8 
60.3 80.4 100.5 | 120.6) 140.7] 160.8 | 180.9 
60.0 §0.0. 100.0 120.0 140.0 160.0 | 180.0 
59.7 79.6 99.5 119.4 139.3 159.2 79.1 
59.4 79.2 99.0 118.8 138.6 158.4 | 178.2 
59.1 78.8 98.5 118.2 137.9 157 Go 1773 
58.8 98.4. 98.0 117.6 1387.2 156.8 | 176.4 
58.5 78.0 97.5 117.0 136.5 156.0 75.5 
58.2 77.6 97.0 |. 116.4} 185.8 | 155.2 |.174.6 
57.9 ite 06.5° |) JIBS") 185.17) 14a eG 
57.6 6.8 96.0 11532 134.4 153.6 | 172.8 
57.3 76.4 O5°5-|- 114.68] 183-9) D520 siie9 
57.0 "6.0 95.0 114.0 133.0 152.0 | 171.0 
56.7 15.6 94.5 | 113.4 | -182.8)] 151.2) 17071 
56.4 15.2 94.0 112.8 131.6 150.4 | 169.2 
56.1 74.8 93.5 112.2 130.9 149.6 | 168.3 
55.8 94.4 93.0} 111.6] 180.2} 148.8 | 167.4 
; 5 55.5 74.0 92.5] 111.0] 129.5 | .148.0 | 166.5 
; oy 55.2 73.6 92.0 110.4 128.8 147.2 |- 165.6 
f ; : 54.9 Woue 91.5 109.8 128.1 146.4 | 164.7 
: ), 54.6 42.8 $1.0 109.2 127.4 145.6 | 163.8 
at f 54.3 92.4 90.5 108.6 126.7 144-8 | 162.9 
y ; F 54.0 72.0 90.0 | 108.0} 126.0] 144.0 | 162.0. 
‘ : 53.7 71.6 89.5 | 107.4] 125.8 | 148.2 | 161.2 
on AERP a a LE IS 


TABLE XXIV.—LOGARITHMS OF NUMBERS 


71 


























































































































No. 240 L. 380.] [No. 269 L. 431. 
N. 0 1 2 8 4 6 6 7 8 9 | Diff. 
240 | 380211 | 0392 | 0578 | 0754 | 0984 |} 1115 | 1296 | 1476 | 1656 | 1837 181 
1 2017 | 2197 | 2377 | 2557 | 2737 || 2917 | 38097 | 3277 | 3456 | 3636 180 
2 3815 | 8995 | 4174 | 4353 | 4533 || 4712 | 4891 | 5070 | 5249 |} 5428 179 
3 5606 | 5785 | 5964 | 6142 | 63821 6499 | 6677 | 6856 ) 7034 | 7212 178 
4 7390 | 7568 | 7746 | 7924 | 8101 || 8279 | 8456 | 8634 | 8811 | 8989 | 178 
5 9166 | 9343 | 9520 | 9698 | 9875 || —— aes SS 
—— = | |} — | —_—_ |__| 0051 | 0228 | 0405 | 0582 | 0759 177 
6 | 390935 | 1112 | 1288 | 1464 | 1641 1817 | 1993 | 2169 | 2845 | 2521 176 
7 2697 | 2873 | 8048 | 3224 | 3400 || 3575 | 3751 | 3926 | 4101 | 4277 176 
8 4452 | 4627 | 4802 | 4977 | 5152 || 5826 | 5501 | 5676 | 5850 | 6025 175 
9 6199 | 6374 | 6548 | 6722 | 6896 || 7071 | 7245 | 7419 | 7592 | 7766 | 174 
250 7940 | 8114 | 8287 | 8461 | 8634 |! 8808 | 8981 | 9154 9328 | 9501 | 173 
1 9674 | 9847 | | Se : 
— 0020 | 0192 | 0365 || 05388 | 0711 | 0883 | 1056 | 1228 7 
> 2} 401401 | 1573 | 1745 | 1917 | 2089 || 2261 | 2488 | 2605 | 2777 | 2949 172 
3 3121 | 3292 | 3464 | 3635 |.3807 || 3978 | 4149 | 4320 | 4492 | 4663 171 
4 4834 | 5005 | 5176 | 5346 | 5517 || 5688 | 5858 | 6029 | 6199 | 6370 171 
5 6540 | 6710 | 6881 | 7051 | 7221 7391 | 7561 | 7731 | 7901 | 8070 170 
6 oa 8410 | 8579 | 8749 | 8918 || 9087 | 9257 | 9426 | 9595 | 9764 | 169 
si 9 — | a 
———| 0102 | 0271 | 0440 | 0609 || 0777 | 0946 | 1114 | 1283 | 1451 | 169 
8 | 411620 | 1788 | 1956 | 2124 | 2298 || 2461 | 2629 | 2796 | 2964 | 3132 168 
9 3300 | 3467 | 3685 | 3803 | 3970 || 4187 | 4805 | 4472 | 4639 | 4806 | 167 
260 4973 | 5140 | 5307 | 5474 | 5641 || 5808 | 5974 | 6141 | 6308 | 6474 | 167 
1 6641 | 6807 | 6973 | 7139 | 7306 || 7472 | 7638 | 7804 | 7970 | 8135 166 
2 po 8467 | 8633 | 8798 | 8964 || 9129 | 9295 | 9460 | 9625 | 9791 | 165 
3 oD ——S$/- ————— | | | —_—_ | ——_. ——- 
—— | 0121 | 0286 | 0451 | 0616 || 0781 | 0945 | 1110 | 1275 | 1439 165 
4 | 421604 | 1768 | 1933 | 2097 | 2261 2426 | 2590 | 2754 | 2918 | 3082 164 
5 3246 | 3410 | 3574 | 3787 | 3901 4065 | 4228 | 4892 | 4555 | 4718 164 
6 4882 | 5045 | 5208 | 5871 | 5534 || 5697 | 5860 | 6023 | 6186 | 6349 163 
lf 6511 | 6674 | 68386 | 6999 | 7161 7324 | 7486 | 7648 | 7811 | 7973 162 
8 8135 | 8297 | 8459 | 8621 | 8783 |} 8944 | 9106 | 9268 | 9429 | 9591 162 
9 Ben 9914 }—— | ———_ |__| ——. 
0286 | 0398 |) 0559 | 0720 | 0881 | 1042 | 1208 161 
PROPORTIONAL PARTS. 
Diff. 1 | 2 3 4 5 6 {C- 8 9 
178 17.8 35.6 53.4 ples 89.0 106.8 124.6 142.4 | 160.2 
Le ete). On. 53.1 70.8 88.5 106.2 | 123.9] 141.6 | 159.3 
ATO 1 -6NT soe 52.8 70.4 88.0 105.6 | 123.2] 140.8 | 158.4 
Lie 17.5 385.0 52.5 70.0 87.5 105.0 pays) 140.0 | 157.5 
174 17.4 34.8 52.2 69.6 87.0 104.4 121.8 139.2 | 156.6 
173 17.3 34.6 51.9 69.2 86.5 108.8 121.1 138.4.| 155.7 
172 pee 34.4 51.6 68.8 86.0 103.2 120.4 137.6 | 154.8 
171 ily test 34.2 ‘lees 68.4 85.5 102.6 119.7 186.8 | 153.9 
170 -.| 17.0 | -34.0 51.0 68.0 85.0 102.0 | 119.0 | 136.0} 153.0 
169 | 16.9| 38.8 50.7 67.6 84.5 101.4 | 118.3] 135.2 | 152.4 
168 | 16.8} 33.6 50.4 67.2 84.0 100.8 | 117.6 | 184.4 | 151.2 
167. | 16:7 | 38.4 5p 50.1 66.8 83.5 100.2} 116.9} 183.6) 150.3 
166 16.6 33.2 49.8 66.4 83.0 99.6 116.2 182.8 | 149.4 
165 16.5 33.0 49.5 66.0 82.5 99.0 115.5 182.0 | 148.5 
164 16.4 32.8 49.2 65.6 82.0 98.4 114.8 181.2 | 147.6 
163 | 16.3} 32.6 48.9 65.2 81.5 97.8 | 114.1}. 180.4 | 146.7 
162 16.2 32.4 48.5 64.8 81.0 97.2 113.4 129.6 | 145.8 
TO1* 7.16.1 4+ 82.2 48.3 64.4 80.5 96.6 |. 112.7 





72 





















































TABLE XXIV.—LOGARITHMS OF NUMBERS 


















































No. 270 L, 431.] [No. 299 L. 476, 
N.| 0 1 28 | 4 | & 18 Diff. 
270 | 431364 | 1525 | 1685 | 1846 | 2007 || 2167 | 2328 161 
1 | 2969 | 8130 | 8290 | 3450 | 3610 || 3770 | 3930 160 
2| 4569 | 4729 | 4888 | 5048 | 5207 || 5367 | 5526 159 
3 | 6163 | 6322 | 6481 | 6640 | 6799 || 6957 | 7116 159 
4| 7751 | 7909 | 8067 | 8226 | 8384 || 8542 | 8701 158 
5 | 9333 | 9491 | 9648 | 9806 | 9964 ||, 
—_——_—|—_—_ ———|—__—|) 0122 | 027 158 
6 | 440909 | 1066 | 1224 | 1381 | 1538 || 1695 | 1852 157 
71 2480 | 2637 | 2793 | 2950 | 3106 || 3263 | 3419 157 
8| 4045 | 4201 | 4357 | 4513 4669 || 4825 | 4981 156 
9} 5604 | 5760 | 5915 | 6071 | 6226 || 6382 | 6537 155 
280 | 7158 | 7313 | 7468 | 7623 | 7778 || 7933 | 8088 155 
1 | 8706 | 8861 | 9015 | 9170 | 9324 || 9478 | 9633 4 
—— ok — Se Sy ae eS 1 
2 | 450249 | 0403 | 0557 | 0711 | 0865 || 1018 | 4472 154 
3| 1786 | 1940 | 2093 | 2247 | 2400 || 2553 | 2706 153 
4} 3318 | 3471 | 3624 | 3777 | 3930 || 4082 | 4235 153 
5 | 4845 | 4997.) 5150 | 5302 | 5454 || 5606 | 5758 152 
6 | 6366 | 6518 | 6670 | 6821 | 6973 || 7125 | 7276 152 
% | 7882 | 8033 | 8184 | 8336 | 8487 || 8638 | 8789 151 
8| 9392 | 9543 | 9694 | 9845 | 9995 || —— 
pe | Se eee ee | ae dee 151 
9 | 460898 | 1048 | 1198 | 1348 | 1499 || 1649 | 1799 150 
290 | 2398 | 2548 | 2697 | 2847 | 2997 ||. 8146 | 8296 150 
1| 3893 | 4042 | 4191 | 4340 | 4490 || 4639 | 4788 149 
2} 5383 | 55382 | 5680 | 5829 | 5977 || 6126 | 6274 149 
3] 6868 | 7016 | 7164 | 7312 | 7460 || 7608 | 7756 148 
4| 8347 | 8495 | 8643 | 8790 | 8938 |} 9085 | 9233 148 
5 | 9822 | 9969 Loeee a 
—_——|——| 0116 | 0263 | 0410 | 0557 | 0704 147 
6 | 471292 | 1438 | 1585 | 1732 | 1878 | 2025 | 2171 146 
7 | 2756 | 2903 | 3049 | 3195 | 8341 || 3487: | 3633 146 
8 | 4216 | 4362 | 4508 | 4653 | 4799 || 4944 | 5090 146 
9| 5671 | 5816 | 5962 | 6107 | 6252 || 6397 | 6542 145 
| 
PROPORTIONAL PARTs, 
Dift.| 1 2 3 4 5 6 7 8 9 
161 | 16.1 | 32.2 | 48.3 | 64.4 | 80.5 | 96.6 | 112.7| 128.8 | 144.9 
160 | 16.0] 32.0 | 48.0 | 64.0 } 80.0 | 96. 112.0} 128.0 | 144.0 
159 | 15.9] 31.8 | 47.7 | 68.6 | 79.5 | 95.4 | 111.3] 127.2 | 143.1 
158 | 15.8] 31.6 | 47.4 | 63.2 ) 79.0 | 94.8 | 110.6] 126.4 | 142.2 
157 | 15.7] 31.4 | 47.1 | 62.8 | 75 | 94.2 | 109.9] 125.6 | 141.3 
156 | 15.6] 31.2 | 46.8 | 62.4 | 78.0 | 93.6 | 109.2] 124.8 | 140.4 
155 | 15.5] 31.0 | 46.5 | 62.0 | 77.5 | 93.0 | 108.5] 124.0 | 139.5 
154 | 15.4] 30.8 | 46.2 | 61.6 7.0 | 92.4 | 107.8 | 123.2 | 138.6 
153 | 15.3) 30.6 | 45.9 | 61.2 | 7.5 | 91.8 | 107.1 | 122.4 | 187.7 
152 | 15.2| 30.4 | 45.6 | 60.8 | 76.0 | 91.2 | 106.4] 121.6 | 136.8 
151 | 15.1} 30:2 | 45.3 | 60.4 | %5.5 | 90.6 | 105.7 | 120.8 | 135.9 
150 | 15.0] 30.0 | 45.0 | 60.0 | %5.0 | 90.0 | 105.0] 120.0 | 185.0 
149 | 14.9] 29.8 | 44.7 | 59.6 | 74.5 | 80.4 | 104.3 | 119.2 | 134.1 
148 | 14.8| 296 | 44.4 | 59.2 | 74.0 | 88.8 | 103.6] 118.4 | 133.2 
147 | 14.7| 29.4 | 44.1 | 58.8 | 73.5 | 88.2 | 102.9] 117.6 | 132.3 
146 | 14.6] 29.2 | 43.8 | 58.4 | 73.0 | 87.6 | 102.2] 116.8 | 131.4 
145 114.5] 29.0 | 43.5 | 58.0 | 72.5 | 87.0 | 101.5] 116.0] 130.5 
144 | 14.4] 28.8 | 48.2 | 57.6 | 72.0 | 86.4 | 100.8] 115.2 | 129.6 
143° | 14.3] 28.6 | 42.9 | 57.2 | 71.5 | 85.8 | 100.1] 114.4 | 128.7 
142 | 14.2) 28.4 | 42.6 | 56.8 | 71.0 85.2 | 99.4] 113.6 | 127.8 
141 |.14.1] 28.2 | 42.3 | 56.4 | 70.5 | 84.6 | 98.7] 112.8 | 126.9 
140 | 14.0] 28.0 | 42.0 | 56.0 | 70.0 | 84.0 98.0 | 112.0 | 126.0 
ar STD 


TABLE XXIV.—LOGARITHMS OF NUMBERS 73 


















































































































































No. 300 L. 477.] (No. 339 L. 531. 
N. G 1 6 6 | 8 9 | Diff. 
800 | 477121 | 7266 | 7411 7844 | 7989 | 8138 | 8278 | 8422 145 
1 8566 | 8711 | 8855 K 9287 | 9431 | 9575 | 9719 | 9863 144 
2 480007 ' 0151 | 0294 725 | 0869 | 1012 | 1156 | 1299 144 
3 14438 | 1586 | 1729 2159 | 2802 | 2445 | 2588 | 2731 143 
4 287 3016 | 3159 38587 | 3780 | 38872 | 4015 | 4157 143 
5 4300 | 4442 ) 4585 5011 | 5158 | 5295 | 5487 | 5579 142 
6 5721 | 5863 | 6005 6480 | 6572 | 6714 | 6855 | 6997 142 
i 7138 | 7280.) 7% 7845 | 7986 | 8127 | 8269 | 8410 141 
3 soe 8692 | 8833 9255 | 9396 | 9587 | 9677 | 9818 141 
5 pene ee |e 
0099 | 0239 0661 | 0801 | 0941 | 1081 | 1222 | 140 
810 | 491362 | 1502 | 1642 2062 | 2201 | 2341 | 2481 | 2621 140 
1 2760 | 2900 | 3040 e 38458 | 3597 | 3737 | 8876 | 4015 139 
ae 4155 | 4294 | 44383 4850 | 4989 | 5128 | 5267 | 5406 189 
3 5544 | 5683 | 5822 6288 | 637 6515 | 6653 } 6791 139 
4 6930 | 7068 | 7 7621 | 7759 | 7897 | 8085 | 8173 138 
5 8311 | 8448 | 8586 8999 | 9137 | 927 9412 | 9550 1388 
6 9687 | 9824 | 9962 es a asa 
—~ 0374 | 0511 | 0648 | 0785 | 0922 | 187 
7.| 501059 | 1196 | 1333 1744 | 1880 | 2017 } 2154 } 2291 137 
B 2427 | 2564 | 2 38109 | 8246 | 8882 | 3518 | 8655 136 
9 3791 | 3927 | 4063 71 | 4607 | 4748 | 4878 | 5014 136 
320 5150 | 5286 | 5421 5828 | 5964 | 6099 | 6234.) 6370 186 
1 6505 | 6640 | 67’ 7181 | 7816 | 7451 | 7586 | 7721 1385 
2 7856 | 7991 | 8126 8530 | 8664 | 8799 | 8984 | 9068 185 
3 9203 | 9837 | 9471 9874. | ——— | | —___ | ___ 
— —. 0009 | 0143 | 0277 | 0411 | 184 
4 | 510545 | 067 08138 1215 | 1849 | 1482 | 1616 |} 1'750 184 
5 1883 | 2017 | 2151 2551 | 2684 | 2818 | 2951 | 38084 183 
6 8218 | 38351 | 3484 8883 | 4016 | 4149 | 4282 | 4415 133 
oy 4548 | 4681 | 4813 5211 | 53844 | 5476 | 5609 | 5741 1388 
8 5874 | 6006 | 6139 65385 | 6668 | 6800 | 69382 | 7064 182 
9 7196 | 7828 | 7 7855 | 7987 | 8119 | 8251 | 8882 | 132 
330 8514 | 8646 | 8 9171 | 9303 | 9484 | 9566 | 9697 | 181 
1| 9828 | 9959 dh ale | alta PR ae iene a 
—_——_}. 0090 0484 | 0615 | 0745 | 0876 | 1007 131 
2 | 521188 | 1269 | 1400 792 | 1922 | 2053 | 2188 | 2314 181 
8| 2444 | 257 ie 3096 | 3226 | 33856 | 38486 | 3616 130 
4 746 | 3887 4006 43896 | 4526 | 4656 | 4785 | 4915 130 
5 5045 | 5174 | 5304 5693 | 5822 | 5951 | 6081 | 6210 129 
6 6339 | 6469 | 6598 6985 | 7114.) 7243 | 737 7501 129 
a 7630 | 7759 | 7888 8274 | 8402 | 8531 | 8660 | 8788} 129 
8 8917 | 9045 | 9174 9559 | 9687 | 9815 | 9943 
ee Se — ———| 0072 128 
9 | 5380200 | 0328 | 0456 0840 | 0968 | 1096 | 1223 | 1351 128 
PROPORTIONAL PARTS. 
Diff 1 2 3 4 5 6 vg 8 9 
139 13.9 27.8 41.7 55.6 69.5 83.4 97.3 2S el Ney l\ et bat | 
138 13.8 27.6 41.4 DD .2 69.0 82.8 96.6 110.4 | 124.2 
137 1327 Sia 41.1 54.8 68.5 82.2 95.9 109.6 | 123.3 
136 13.6 af Ger | 40.8 54.4 68.0 81.6 95.2 108.8 | 122.4 
135 13.5 27.0° 40.5 54.0 67.5 81.0 94.5 108.0 | 121.5 
134 13.4 26.8 40.2 53.6 67.0 80.4 93.8 107.2 | 120.6 
133 .| 18.3 | - 26.6 39.9 53.2 66.5 79.8 93.1 106.4 | 119.7 
132 13.2 26.4 39.6 52.8 66.0 79.2 92.4 105.6 | 118.8 
Aslerietoel ) eb. 89.3 52.4 65.5 "8.6 91.7 104.8 | 117.9 
130 13.0 | .26.0 39.0 52.0 65.0 78.0 91.0 104.0 | 117.0 
129 12.9 | 25.8 38.7 51.6 64.5 77.4 90.3 103.2 | 116.1 
128 LESH] ed. 6 88.4 51.2 64.0 76.8 89.6 102.4 | 115.2 
127 eee 2574 38.1 50.8 63.5 76.2 88.9 101.6 } 114.2 





74 TABLE XXIV.—LOGARITHMS OF NUMBERS 


LL LLL ED 
























































No, 340 L, 531.] [No. 879 L. 579. 
N. 0 1 2 B 4 5 6 | 7 8 9 | Diff. 
340 | 538147 1607 | 1734 | 1862 | 1990 || 2117 | 2245 | 2372 | 2500 | 2627 128 
1 27 2882 { 3009 | 8136 | 8264 || 3391 ; 3518 | 3645 | 8772 | 3899 127 
2 4026 | 4153 | 4280 | 4407 | 45384 || 4661 | 4787 | 4914 | 5041 | 5167 127 
3 5204 | 5421 | 5547 | 5674 | 5800 || 5927 | 6053 | 6180 306 | 6482 126 
4 6558 | 6685 | 6811 | 6937 | 7063 || 7189 | 7315 | 7441 | 7567 | 7693 126 
5 7819 | 7945 | 8071 | 8197 | 8322 || 8448 | 8574 | 8699 | 8825 | 8951 126 
6 907 9202 | 9327 | 9452 | 9578 || 9703 | 9829 | 9954 
——_— | ——_ ——}| 007 0204 125 
"| 540329 | 0455 | 0580 | 0705 | 0830 || 0955 | 1080 | 1205 | 1830 | 1454 125 
8 1579 | 1704 | 1829 | 1953 | 2078 || 2203 | 2827 | 2452 | 257 2701 125 
9 | 2825 | 2950 | 3074 | 8199 | 3323 || 3447 | 3571 | 8696 | 3820 | 3944 | 124 
350 4068 | 4192 | 4316 | 4440 | 4564 || 4688 | 4812 | 4936 | 5060 | 5183 124 
1 5307 | 5431 | 5555 | 5678 | 5802 || 5925 | 6049 | 6172 | 6296 | 6419 124 
2 6543 | 6666 | 6789 | 6913 | 7036 || 7159 | 7282 | 7405 | 7529 | 7652 123 
3 “775 | 7898 | 8021 | 8144 | 8267 || 8389 | 8512 | 8635 | 8758 | 8881 123 
4 9003 | 9126 | 9249 | 9371 | 9494 || 9616 | 9739 } 9861 | 9984 Ga = 
| | | | 0 
5 | 550228 | 0351 | 0473 | 0595 | 0717 || 0840 | 0962 | 1084 | 1206 | 1328 122 


1450 | 1572 | 1694 | 1816 | 1938 || 2060 | 2181 | 2803 | 2425 | 2547 | 122 


to 
S 











| | | 





561104 | 1221 | 1340 | 1459 | 1578 || 1698 | 1817 | 1986 | 2055 | 2174 | 119 
2293 | 2412 | 2531 | 2650 | 2769 || 2887 | 8006 | 3125 | 3244 | 3362 | 119 
3481 | 3600 | 3718 | 3837 | 3955 || 4074 | 4192 | 4311 | 4429 | 4548] 119 
4666 | 4784 | 4903 | 5021 | 5139 257 | 53876 | 5494 | 5612 | 5730 | 118 
5848 | 5966 | 6084 | 6202 | 6320 || 6437 | 6555 | 6673 | 6791 | 6909 | 118 
7026 | 7144 | 7262 | 7879 | 7497 || 7614 | 7732 | 7849 | 7967 | 8084 | 118 


8202 | 8319 | 8436 | 8554 | 8671 || 8788 | 8905.| 9023 | 9140 | 9257 | 117 
9374 | 9491 | 9608 | 9725 | 9842 || 9959 


570543 | 0660 | 0776 | 0893 | 1010 || 1126 | 1243 | 1859 | 1476 | 1592 | 117 
1709 | 1825 | 1942 | 2058 | 2174 || 2291 | 2407 | 2523 | 2639 | 2755] 116 
2872 | 2988 | 8104 | 3220 | 8336 || 8452 | 8568 | 3684 | 3800 | 3915 | 116 
4031 | 4147 | 4263 | 4379 | 4494 || 4610 | 4726 | 4841 | 4957 | 5072 | 116 
5188 | 5303 | 5419 |.5534 | 5650 || 5765 | 5880 | 5996 | 6111 | 6226 | 115 
6341 | 6457 | 6572 | 6687 | 6802 || 6917 | 70382 | 7147 | 7262 | 7377 | 115 
7492 | 7607 | 7722 | 7836 | 7951 |) 8066 | 8181 | 8295 | 8410 | 8525 | 115 
8639 | 8754 | 8868 | 8983 | 9097 || 9212 | 93826 | 9441 | 9555 | 9669 | 114 








o9 
COMO RwWW HS COVOom WwWHS wHAID 























Diff 1 2 3 | 4 | 5 | 6 if | 8 | 9 

128 | 12.8) 25.6 388.4 51.2 64.0 76.8 89.6 | 102.4 | 115.2 
127 | 12.7 | 25.4 38.1 50.8 63.5 76.2 88.9 | 101.6 | 114.3 
126 | 12.6] 25.2 37.8 50.4 63.0 75.6 88.2 | 100.8 | 113.4 
125. | 12.5 | 25.0 37.5 50.0 62.5 75.0 87.5 | 100.0 | 112.5 
124 | 12.4) 24.8 37.2 49.6 62 74.4 86.8 99.2 | 111.6 
123 | 12.3] 24.6 36.9 49.2 61.5 73.8 86.1 98.4 | 110.7 
122.1122) 24.4 36.6 48.8 61.0 73.2 85.4 97.6 | 109.8 
121. | 12.1} 24.2 36.3 48.4 60.5 72.6 84.7 96.8 | 108.9 
120 }12.0|] 24.0 36.0 48.0 60.0 72.0 84.0 96.0 | 108.0 
119: | 11.9] 23.8 35.7 47.6 59.5 71.4 83.3 95.2 | 107.1 





‘ TABLE. XXIV.—LOGARITHMS OF NUMBERS 75 










































































No. 380. 1, 579.) [No. 414 L. 617. 
“geal ibeeel se oS Mh Pci, 
BOO | SOTHe | GENs 0012 | 0126 0241 || 0355 | 0469 | 0583 | 0697 | 0811 | 114 
1 | 580925 | 1039 | 1153 { 1267 | 1381 || 1495 | 1608 | 1722 | 1886 | 1950 
2/ 2063 | 2177 | 2291 | 2404 | 2518 || 2631 | 2745 | 2858 | 2972 | 3085 
8] | 3199 | 33842 | 3426 | 3539 | 3652 || 3765 | 3879 | 3992 | 4105 | 4218 
4| 4331 | 4444 | 4557 | 4670 | 4788 || 4896 | 5009 | 5122 | 5235 | 5348] 113 
5 | 5461 | 5574 | 5686 | 5799 | 5912 || 6024 | 6137 | 6250 | 6362 | 6475 
6 | 6587 | 6700 | 6812 | 6925 | 7037 || 7149 | 7262 | 7374 | 7486 | 7599 
71 ‘711 | 7823 | 7935 | 8047 | 8160 || 8272 | 8384 | 8496 | 8608 | 8720 | 112 
8 oe 8944 | 9056 | 9167 | 9279 || 9391 | 9503 | 9615 | 9726 | 9838 
————| 0061 | 0173 | 0284 | 0396 |) 0507 | 0619 | 0730 | 0842 | 0953 
890 | 591065 | 1176 | 1287 | 1399 | 1510 || 1621 | 1732 | 1843 | 1955 | 2066 
.1 | 2177 | 2288 | 2399 | 2510 | 2621 || 2732 | 2843 | 2954 | 3064 | 3175 | 114 
2| 3286] 3397 | 3508 | 3618 | 3729 || 3840 | 3950 | 4061 | 4171 | 4282 
3| 4893] 4503 | 4614 | 4724 | 4884 || 4945 | 5055 | 5165 | 5276 | 5386 
4| 5496 | 5606 | 5717 | 5827 | 5937 || 6047 | 6157 | 6267 | 6377 | 6487 
5 | 6597 | evoy | 6817 | 6927 | 70387 || 7146 | 7256 | 7366 | 7476 | 7586 | 110 
6| 7695 | 7805 | 7914 | 8024 | 8134 || 8243 | 8353 | 8462 | 8572 | 8681 
71 8791 | 8900 | 9009 | 9119 | 9228 || 9337 | 9446 | 9556 | 9665 | 977 
Rijn Ong eee) le ee ee ee he 109 
—~+—|—~—] 0101 | 0210 | 0319 || 0428 | 0537 | 0646 | 0755 | 0864 
9 | 600973 | 1082 |. 1191 | 1299 | 1408 || 1517 | 1625 | 1734 | 1843 | 1951 
400} 2060 | 2169 | 2277 | 2386 | 2494 || 2603 | 2711 | 2819 | 2928 | 3036 
1| 38144 | 3253 | 3361 | 3469 | 3577 |) 3686 | 3794 | 8902 | 4010 | 4118] 408 
2| 4226 | 4334 | 4442 | 4550 | 4658 || 4766 | 4874 | 4982 | 5089 | 5197 
3 5413 | 5521 | 5628 | 5736 || 5844 | 5951 | 6059 | 6166 | 6274 
4 6489 | 6596 | 6704 | 6811 || 6919 | 7026 | 7133 | 7241 | 7348 
5 7562 | 7669 | 7777 | 7884 || 7991 | 8098 | 8205 | 8312 | 8419] 407 
6 8633 | 8740 | 8847 | 8954 || 9061 | 9167 | 9274 | 9881 | 9488 
By 9701 | 9808 | 9914 een 
0021 || 0128 | 0284 | 0841 | 0447 | 0554 
8 0767 | 0873 | 0979 | 1086 || 1192 | 1298 | 1405 | 1511 | 1617 
9 1829 | 1936 | 2042 | 2148 || 2254 | 2360 | 2466 | 2572 | 2678 | 40g 
410 2890 | 2996 | 3102 | 3207 || 3313 | 3419 | 3525 | 3630 | 3736 
1 3947 | 4053 | 4159 | 4264 || 4370 | 4475 | 4581 | 4686 | 4792 
2 5003 | 5108 | 5213 | 5319 || 5424 | 5529 | 5634 | 5740 | 5845 
3 6055 | 6160 | 6265 | 6870 || 6476 | 6581 | 6686 | 6790 | 6895 | 105 
4 7105 | 7210 | 7815 | 7420 || 7525 | 7629 | 7734 











7839 | 7948 














Diff.| 1 2 3 4 | 5 | 6 Y 8 9 

118 11.8 23.6 35.4 ee, 59.0 70.8 82.6 94.4 106.2 
117 |11.7| 93.4 | 35.1 | 46.8 | 58.5 | 70.2 | 81.9 | 93.6 | 105.3 
116 |11.6| 23.2 | 24.8 | 46.4 | 58.0 | 69.6.| 81.2 | 92.8 | 104.4 
115 | 11:5| 23.0 | 34.5 | 46.0 | 57.5 | 69.0 | 80.5 | 92.0 | 108.5 
114. [-11.4'| 93:8 .} 34.2 456 7.0. | 68.4 | 79.8 |.91.2 | 102.6 
113 -|.11-3 }- 29-6 }.-33.9 “| 45-2" | -56.5-| > 67.8 1 -79.1-1- 90.4 | 101-7 
112 | 11.2| 22.4 | 338.6 | 44.8 | 56.0 | 67.2 | 78.4 | 89.6 | 100.8 
111 | 11.1] 22.2 | 33.3 | 44.4 | 55.5 | 66.6 | 77.7 | 88.8 | 99.9 
110 | 11.0| 22.0 1° 33.0 | 44.0] 55.0 | 66.0 | 77.0 | 88.0 | 99.0 
109 | 10.9| 21.8 | 32-7 | 43:6 | 54.5 -| 65.4 | 76.3 7.2 | 98.1 
108 |10.8| 21.6 | 32.4 | 43.2 | 54.0 | 64.8 | 75.6 | 86.4 | 97.2 
107 | 10.7] 21.4 | 32.1 | 42.8 | 53.5 | 64.2 | 74.9 | 85.6 { 96.3 
106. | 10.6} 21.2 | 31.8 | 42.4 | 58.0 | 63.6 | 74.2 | 84.8 | 95.4 
105 | 10.51 21.0 | 31.5 | 42.0 | 52.5 | 63.0 | 73.5 | 84.0 | 94.5 
105 110.5 | 21.0 | 31.5 | 42.0 | 525 | 63.0 | v5 | g4.0 | 94.5 
104 | 10.4] 20.8 | 31.2 | 41.6 | 52.0 | 62.4 | v2'8 | 93.2 | 93.6 








76 


TABLE XXIV.—LOGARITHMS 


OF NUMBERS 





No. 415 L. 618.] 































































































[No, 459 L. 662, 




















N. 0 1 2 3 4 5 6 a 8 9 Diff. 
415 | 618048 | 8153 | 8257 | 8362 | 8466 || 8571 | 8676 | 8780 | 8884 | 8989 105 
6 9093 | 9198 | 9802 | 9406 | 9511 9615 | 9719 | 9824 | 9928 |——_— 
a ——_—— | —— || ——- | -—_— | _ - —-—-} 0082 
7 | 620136 | 0240 | 0344 | 0448 | 0552 || 0656 | 0760 | 0864 | 0968 | 107 104 
8 1176 | 1280 | 13884 | 1488 | 1592 || 1695 | 1799 | 1903 | 2007 | 2110 
9 2214 | 2318 | 2421 | 2525 | 2628 || 2782 | 2885 | 2989 | 8042 | 3146 
420 8249 | 3353 | 8456 | 38559 | 3663 || 3766 | 38869 | 3973 | 4076 | 417 
1 4282 | 4385 | 4488 | 4591 | 4695 || 4798 | 4901 | 5004 | 5107 | 5210 108 
rm 5312 | 5415 | 5518 | 5621 | 5724 || 5827 | 5929 | 6032 | 6135 | 6238 
3 6340 | 6448 | 6546 | 6648 | 6751 6853 | 6956 | 7058 | 7161 | 7263 
4 7366 | 7468 | 7571 | 76738 | 7775 || 7878 | 7980 | 8082 | 8185 | 8287 
5 8389 | 8491 | 8593 | 8695 | 8797 || 8900 | 9002 | 9104 | 9206 | 9308 102 
6 | * 9410 | 9512 | 9613 | 9715 | 9817 9919 |—————_ ——_—_ | —__- 
——_— | —_—- | —_—_ | —____ |__| |__| 0021 | 01238 | 0224 | 0826 
” | 680428 | 0530 | 0631 | 0788 | 0835 || 0936 | 1088 | 1139 | 1241 | 1342 
8 1444 | 1545 | 1647 | 1748 | 1849 1951 (1 2052 | 2158 f 22bp 2806 
9 2457 | 2559 | 2660 | 2761 | 2862 || 2963 | 8064 | 3165 | 8266 | 3367 
430 3468 | 3569 | 3670 | 8771 | 3872 || 3973 | 4074 | 4175 | 4276 | 4876 | 101 
1 4477 | 4578 | 4679 | 4779 | 4880 || 4981 | 5081 | 5182 | 5283 | 5883 
2 5484 | 5584 | 5685 | 5785 | 5886 || 5986 | 6087 | 6187 | 6287 | 63888 
3 6488 | 6588 | 6688 | 6789 | 6889 || 6989 | 7089 | 7189 | 7290 | 7390 
4 7490 | 7590 | 7690 | 7790 | 7890 || 7990 | 8090 | 8190 | 8290 | 8889 100 
5 8489 | 8589 | 8689 | 8789 | 8888 |} 8988 | 9088 | 9188 | 9287 | 9387 
6 9486 | 9586 | 9686 | 9785 | 9885 || 9984 |——/ —_—_| —_—__ | __—__ 
ee ee a — —| 0084 | 0183 | 0288 | 0382 
Y | 640481 |} 0581 | 0680 | 077 087 0978 | 107 1177 | 12% 1875 
8 147 157 1672 | 1771 | 187 197 2069 | 2168 | 2267 | 23866 
9 2465 | 2563 | 2662 | 2761 | 2860 || 2959 | 8058 | 3156 | 38255 | 3354 99 
440 3453 | 3551 | 3650 | 8749 | 8847 || 3946 | 4044 | 4143 | 4242 | 4340 
1 4489 | 45387 | 4686 | 47. 4832 || 4931 | 5029 | 5127 | 5226 | 5824 
2 5422 | 5521 | 5619 | 5717 | 5815 || 5913 | 6011 | 6110 | 6208 | 6306 
3 6404 | 6502 | 6600 | 6698 | 6796 || 6894 | 6992 | 7089 | 7187 | 7285 98 
4 7383 | 7481 | 7579 | 7676 | 777: 7872 | 7969 | 8067 | 8165 | 8262 
5 8860 | 8458 | 8555 | 8653 | 8750 || 8848 | 8945 |.9043 | 9140 | 9237 
6 938385 | 9482 | 9580 | 9627 | 9724 || 9821 | 9919 |——_—-|—__—_|_ — 
| ——} | | 0016 } Ota 0210 
7 | 650308 | 0405 | 0502 | 0599 | 0696 || 0793 | 0890 | 0987 | 1084 | 1181 
8 1278 } 1375 | 1472 | 1569 | 1666 || 1762 | 1859 | 1956 | 2058 | 2150 97 
9 2246 | 2843 | 2440 | 2586 | 2683 || 2780 | 2826 | 2923 | 3019 | 3116 
450 8213 | 3309 | 3405 | 3502 | 3598 || 8695 | 3791 | 3888 | 38984 | 4080 
1 4177 | 4278 | 4869 | 4465 | 4562 |; 4658 754 | 4850 | 4946 | 5042 
2 5188 | 5235 | 5831 | 5427 | 5528 || 5619 | 5715 | 5810 | 5906 | 6002 96 
3 6098 | 6194 | 6290 | 6386 | 6482 || 6577 | 667 6769 | 6864 | 6960 
4 7056 | 7152 | 7247 | 7843 | 7488 || 7584 | 7629 | 7725 | 7820 | 7916 
5 8011 | 8107 | 8202 | 8298 | 8393 || 8488 | 8584 | 8679 | 8774 | 8870 
: ae 9060 | 9155 | 9250 | 9346 9441 | 9586 | 9631 726 | 9821 
— ——} 0011 | 0106 | 0201 | 0296 || 0391 | 0486 | 0581 | 0676 | O771 95 
8 | 660865 | 0960 | 1055 | 1150 | 1245 || 1389 | 1484 | 1529 | 1623 | 1718 
9 1813 | 1907 | 2002 | 2096 j 2191 || 2286 | 2380 | 2475 | 2569 | 2668 
PROPORTIONAL PARTS, 
Diff iL 2 3 4 5 6 q 8 9 
105 | 10.5] 21.0 81.5 42.0 52.5 63.0 73.5 84.0 : 
104 | 10.4] 20.8 31/2 41.6 52.0 62.4 72 8 83.2 bee 
1038 10.3 20.6 30.9 41.2 51.5 61.8 oo 82.4 92.7 
1022 10.2% 20.4) \] S060). 4084]. 5120s)| 61iSeal Serette! Ber-6! | fonts 
101 | 10.1 | 20.2 | 30.3 | 40.4 | 50.5 | 60.6 | 707 | 80.8 | .90.9 
100 | 10.0] 20.0 30.0 40.0 50.0 60.0 70 0 80.0 90.0 
99 | 9.9] 19.8 | 29.7 | 89.6 | 49.5 | 59.4 | 69.3 | 79.2 | 89.1 
—_——— LAS 


TABLE XXIV.—LOGARITHMS OF NUMBERS 77 























































































































No. 460 L. 662.] [No. 499 L. 698, 
N. 0 1 2 § 4 5 6 7 | 8 9 Diff. 
460 | 662758 | 2852 | 2947 | 3041 | 3135 || 3230 | 3824 | 8418 | 3512 | 38607 
1 8701 | 3795 | 8889 | 8988 | 407 4172 | 4266 | 4860 | 4454 | 4548 
2 4642 736 | 4880 | 4924 | 5018 || 5112 | 5206 | 5299 | 5398 | 5487 94 
3 5581 | 567 5769 | 5862 | 5956 || 6050 | 6148 | 6287 | 6331 | 6424 
4 6518 | 6612 | 6705 | 6799 | 6892 || 6986 | 707 7173 | 7266 | 7360 
5 7453 | 7546 | 7640 | 7733 | 7826 || 7920 | 8013 | 8106 | 8199 | 8293 
6 8386 | 847’ 8572 | 8665 | 8759 || 8852 | 8945 | 9088 | 9131 | 9224 
fi 9317 | 9410 | 9503 | 9596 | 9689 || 9782 | 9875 | 9967 |-——-|_——_ 
Ben | ————| —___— || | 1OODO I OLDS 93 
8 | 670246 | 0839 | 0431 | 0524 | 0617 || 0710 | 0802 | 0895 | 0988 | 1080 
9 1173 | 1265 | 1858 | 1451 | 1543 || 1686 | 1728 | 1821 | 1913 | 2005 

470 2098 | 2190 | 2283 | 2375 | 2467 || 2560 | 2652 | 2744 | 2886 | 2929 
1 8021 | 8113 | 3205 | 3297 | 33890 || 3482 | 3574 | 8666 | 38758 | 3850 
2 8942 | 4034 | 4126 | 4218 | 4810 || 4402 | 4494 | 4586 | 467 4769 1 92 

ts} 4861 | 4953 | 5045 | 5187 | 5228 || 53820 | 5412 | 5503 | 5595 | 5687 
4 5778 | 587 5962 | 6053 | 6145 || 6236 | 6328 | 6419 | 6511 | 6602 
5 6694 | 6785 | 6876 | 6968 |} 7059 |) 7151 | 7242 | 7333 | 7424 | 7516 
6 7607 |. 7698 | 7789 | 7881 | 7972 || 8063 | 8154 | 8245 4 83836 | 8427 
q 8518 | 8609 | 8700 | 8791 | 8882 || 8973 | 9064 | 9155 | 9246 | 9337 91 
8 9428 | 9519 | 9610 | 9700 | 9791 |; 9882 | 9973 Se 

——_—___— 0063 | 0154 | 0245 
9 | 680336 | 0426 | 0517 | 0607 | 0698 |; 0789 | 0879 | 0970 | 1060 | 1151 

480 1241 | 13382 | 1422 | 1513 | 1603 || 1693 | 1784 | 1874 | 1964 | 2055 
1 2145 | 2235 | 2826 | 2416 | 2506 || 2596 | 2686 | 2777 | 2867 | 2957 
2 3047 | 3187 | 8227 | 3817 | 3407 || 8497 | 3587 | 8677 | 38767 | 3857 90 
3 3947 | 40387 | 4127 | 4217 | 4307 || 4896 | 4486 | 4576 | 4666 | 4756 
4 4845 | 49385 | 5025 | 5114 | 5204 || 5294 | 5383 | 5473 | 5568 | 5652 
5 5742 | 5831 | 5921 | 6010 | 6100 || 6189 | 6279 | 6368 | 6458 | 6547 
6 6636 | 6726 | 6815 | 6904 | 6994 || 7083 | 7172 | 7261 | 7351 | 7440 
7h 7529 | 7618 | 7707 | 7796 | 7886 || 797 8064 | 8153 | 8242 | 8831 89 
8 8420 | 8509 | 8598 | 8687 | 8776 || 8865 | 8953 | 9042 | 91381 | 9220 

- 9 9309 | 9398 | 9486 | 957 9664 || 9753 | 9841 | 9930 |——— 

Eee. Sake 3 ere te el ee MIE: IOI Cy. 

490 | 690196 | 0285 | 0373 | 0462 | 0550 || 0639 | 0728 | 0816 | 0905 | 0993 
1 1081 | 1170 | 1258 | 1847 | 1435 || 1524 | 1612 | 1700 | 1789 | 1877 
2 1965 | 2053 | 2142 | 2280 | 2318 9406 | 2494 | 2588 | 2671 | 2759 
3 2847 | 2985 | 3023 | 8111 | 3199 || 8287 | 3375 | 8463 | 38551 | 38639 88 
4 38727 | 3815 | 3903 | 8991 | 407 4166 | 4254 | 4342 | 4480 | 4517 
5 4605 | 4693 | 4781 | 4868 | 4956 5044 | 5131 | 5219 | 53807 | 5394 
6 5482 | 5589 | 5657 | 5744 | 5882 || 5919 | 6007 | 6094 | 6182 | 6269 
% 6356 6531 | 6618 | 6706 || 6793 | 6880 | 6968 | 7055 | 7142 
8 7229 | 7317 | 7404 | 7491 | 7578 || 7665 | 7752 | 7839 | 7926 | 8014 8 
9 8100 | 8188 | 8275 | 8862 | 8449 || 8535 | 8622 | 8709 | 8796 | 8883 7 

PROPORTIONAL PARTS. 

Diff 1 2 | 3 4 5 6 v4 8 9 
98 9.8 19.6 29.4 89.2 49.0 58.8 68.6 78.4 88.2 
97 9.7 19.4 29.1 38.8 48.5 58.2 67.9 "7.6 87.3 
96 9.6 19.2 28.8 88.4 48.0 57.6 67.2 76.8 86.4 
95 9.5 19.0 4, 28.5 88.0 47.5 57.0 66.5 76.0 85.5 
94 9.4] 18.8 28.2 37.6 47.0 56.4 65.8 75.2 84.6 
93 9.3 | 18.6 27.9 37.2 46.5 55.8 65.1 74.4 83.7 
92 9.2 18.4 27.6 86.8 46.0 a 64.4 73.6 82.8. 
91 914). 18.2 27.3 36.4 45.5 54.6 63.7 72.8 81.9 
90 9.0 18.0 27.0 36.0 45.0 54.0 63.0 | .72.0 81.0 
89 8.9] 17.8 26.7 85.6 44.5 53.4 62.3 71.2 80.1 
88 8.8] 17.6 26.4 85.2 44.0 52.8 61.6 70.4 79.2 
87 8.7 17.4 26.1 84.8 43.5 52.2 60.9 69.6 Wtert>) 
86 8.6 | 17.2 25.8 34,4 43.0 51.6 60.2 68.8 77.4 

1 neem aaa aaah 


78 TABLE XXIV.—LOGARITHMS OF NUMBERS 


































































































No, 500 L, 698. [No. 544 L. 786. 
| 
Mey 0 “Wee aot Boker} F3 6°18 18 9 | Diff. 
500 | 698970 | 9057 | 9144 | 9231 | 9817 || 9404 | 9491 | 9578 | 9664 | 9751 
1} "eaee |) gga |e | Se | |e | Se 
\——_—— | 0011 | 0098 | 0184 || 0271 | 0358 | 0444 | 0531 | 0617 
2 | 700704 | 0790 | 0877 | 0963 | 1050 || 1136 | 1222 | 1809 | 1395 | 1482 
3 | 1568 | 1654 | 1741 | 1827 | 1913 || 1999 | 2086 | 2172 | 2258 | 2844 
4| 2431 | 2517 | 2603 | 2689 | 2775 || 2861 | 2947 | 8038 | 3119 | 3205 
5 | 3291 | 3377 | 8463 | 3549 | 3635 || 38721 | 8807 | 8893 | 3979 | 4065 | 86 
6 | 4151 | 4236 | 4322 | 4408 | 4494 || 4579 | 4665 | 4751 | 4887 | 4922 
7 | 5008 | 5094 | 5179 | 5265 | 5350 || 5436 | 5522 | 5607 | 5693 | 5778 
8 | 5864 | 5949 | 6035 | 6120 | 6206 || 6291 | 6376 | 6462 | 6547 | 6632 
9 | 6718 | 6803 | 6888 | 6974 | 7059 || 7144 | 7229 | 7315 | 7400 | 7485 
510 | 7570 | 7655 | 7740 | 7826 | 7911 || 7996 | 8081 | 8166 | 8251 | 8336 | gx 
1 | 8421 | 8506 | 8591 | 8676 | 8761 || 8846 | 8931 | 9015 | 9100 | 9185 
2 | 9270 | 9355 | 9440 | 924 | 9609 || 9604 | 9770 | 9863 | 9948 ——— 
3 | 710117 | 0202 | 0287 | 0871 | 0456 || 0540 | 0625 | 0710 | 0794 | o879 
4| 0963 | 1048 | 1132 | 1217 | 1301 || 1385 |.1470 | 1554 | 1639 | 1723 
5} 1807 | 1892 | 1976 | 2060 | 2144 || 2229 | 2318 | 2397 | 2481 | 2566 
6 | 2650 | 2734 | 2818 | 2902 | 2086 || 8070 | 8154 | 3238 | 8328 | 3407 | gy 
7 | 3491 | 8575 | 3659 | 3742 | 3826 || 3910 | 3994 | 4078 | 4162 | 4246 
8| 4330 | 4414 | 4497 | 4581 | 4665 || 4749 | 4883 | 4916 | 5000 | 5084 
9| 5167 | 5251 | 5385 | 5418 | 5502 || 5586 | 5669 | 5753 | 5836 | 5920 
520 | 6003 | 6087 | 6170 | 6254 | 6837 || 6421 | 6504 | 6588 | 6671 | 6754 
1| 6888 | 6921 | 7004.| 7088 | 7171 || 7254 | 7338 | 7421 | 7504 | 7587 
2| V6v1 | W754 | 7837 | 7920 | 8003 || 8086 | 8169 | 8253 | 8336 | 8119 | go 
3| 8502 | 8585 | 8668 | 8751 | 8834 || 8917 | 9000 | 9083 | 9165 | 9248 
4| 9331 | 9414 | 9497 | 9580 | 9663 || 9745 | 9828 | 9911 | 9994 aii 
B | 720159 | 0242 | 0325 | 0407 | 0490 || 0573 | 0655 | 0738 | 0821 | 0903 
6 | 0986 | 1068 | 1151 | 1233 | 1316 || 1398 | 1481 | 1563 | 1646 | 1728 
7 | 1811 | 1893 | 1975 | 2058 | 2140 || 2292 | 2305 | 2387 | 2469 | 2552 
8) 2634 | 2716 | 2798 | 2881 | 2963 || 3045 | 3127 | 3209 | 3291 | 3374 
9) 3456 | 3538 | 3620 | 8702 | 3784 || 8866 | 3948 | 4030 | 4112 | 4194 | 82 
580 | 4276 | 4858 | 4440 | 4522 | 4604 || 4685 | 4767 | 4849 | 4931 | 5013 
1| 5095 | 5176 | 5258 | 5840 | 5422 || 5508 | 5585 | 5667 | 5748 | 5830 
2/| 5912 | 5993 | 6075 | 6156 | 6238 || 6320 | 6401 | 6483 | 6564 | 6646 
3 | 6727 | 6809 | 6890 | 6972 | 7053 || 7134 | 7216 | 7297 | 7379 | 7460 
4| 541 | 7623 | 7704 | 7785 | 7866 || 7948 | 8029 | 8110 | 8191 | 827 
5 | 8354 | 8435 | 8516 | 8597 | 8678 || 8759 | 8841 | 8922 | 9003 | 9084 
é 165 | 9246 | 9327 9408 | 9489 || 9570 | 9651 | 9732 | 9818 | 9893 | 81 
Fe EIT AN RE | Pe eS A | ee ae 2 ee Se Bees 2B Be ae 2 
—| 0055 | 0136 | 0217 | 0298 || 0378 | 0459 | 0540 | 0621 | 0702 
8 | 730782 | 0863 | 0944 | 1024 | 1105 || 1186 | 1266 | 1347 | 1428 | 1508 
9 | 1589 | 1669 | 1750 | 1830 | 1911 || 1991 | 2072 | 2152 | 2238 | 2313 
B40 | 2394 | 2474 | 2555 | 2685 | 2715 || 2796 | 2876 | 2956 | 3037 | 3117 
1| 38197 | 3278 | 3358 | 3438 | 8518 || 3598 | 3679 | 3759 | 3839 | 3919 
2| 3999 | 4079 | 4160 | 4240 | 4320 || 4400 | 4480 | 4560 | 4640 | 4720 | gy 
3 4800 | 4880 | 4960 | 5040 | 5120 || 5200 | 5279 | 5359 | 5439 | 5519 
4| 5599 | 5679 | 5759 | 5838 | 5918 || 5998 | 6078 | 6157 | 6237 | 6317 




















TABLE XXIV.—LOGARITHMS OF NUMBERS 79 












































































































































No. 545 L. 736.] [No. 584 L. 767. 

oN, 0 1 2 3 4 5 6 vi 8 9 | Diff. 

545 | 786397 | 6476 | 6556 | 6635 | 6715 || 6795 | 6874 | 6954 | 7034 | 71138 
6 7193 | 7272 | 7852 | 7431 | 7511 7590 | 7670 | 7749 | 7829 | 7908 
fi 7987 | 8067 | 8146 | 8225 | 8305 || 8384 | 8463 | 8543 | 8622 | 8701 
8 8781 | 8860 | 8939 | 9018 | 9097 || 9177 | 9256 | 9335 | 9414 | 9498 
9 Ga? 1960101 97a, |. 9810. || 988001. 9968, |——_—=— | | re 

$$ | | | —_— 0047 | 0126 | 0205 | 0284 "9 

550 | 740368 | 0442 | 0521 | 0600 | 0678 || O757 | 0886 | 0915 | 0994 | 1073 
1 1152 | 1280 | 1809 | 1888 | 1467 1546 | 1624 | 1703 | 1782 | 1860 
2 1939 | 2018 | 2096 | 2175 | 2254 || 2382 | 2411 | 2489 | 2568 | 2647 
54 2725 | 2804 | 2882 | 2961 | 8089 || 3118 | 3196 | 3275 | 8353 | 3431 
4! 3510 | 3588 | 3667 | 3745 | 3823 ||-3902 | 3980 | 4058 | 4136 | 4215 
5 ' 42938 | 4871 | 4449 | 4528 | 4606 || 4684 | 4762 | 4840 | 4919 | 4997 
6 5075 | 5153 | 5281 | 5809 | 5887 || 5465 | 5543 | 5621 | 5699 | 5777 "8 
fi 5855 | 59383 6011 | 6089 | 6167 || 6245 ' 6323 | 6401 | 6479 | 6556 
8 6634 712 | 6790 | 6868 | 6945 70238 7101 | 7179 | 7256 | 7334 
9 | TAI | 7489 | 1567 | 7645 | 7722 || 7800 7878 | 7955 | 8033 | 8110 

560 8188 | 8266 | 8343 | 8421 | 8498 || 8576 8653 | 8731 | 8808 | 8885 
i 8963 | .9040 | 9118 | 9195 | 9272 9350 9427 | 9504 | 9582 | 9659 
2 9736 | 9814 | 9891 | 9968 |———| |—___ —__ 

—- ——_—_—. —— | 0045 |} 0123 0200 | 0277 | 03854 | 0431 
8 | 750508 | 0586 | 0663 | 0740 | 0817 0894 0971 | 1048 | 1125 | 1202 
4 127 1356 | 1433 | 1510 | 1587 1664 1741 | 1818 | 1895 | 1972 we 
5 2048 | 2125. | 2202 | 2279 | 2356 2483 2509 | 2586 | 2668 | 2740 
6 2816 | 2893 | 2970 | 8047 | 3128 || 8200 8277 | 38853 | 3430 | 3506 
7 3583 | 3660 | 3736 | 3813 | 38889 8966 4042 | 4119 | 4195 | 427 
8 4348 | 4425 | 4501 | 4578 | 4654 || 4730 4807 | 4888 | 4960 | 5036 
9 5112 | 5189 | 5265 | 5841 | 5417 || 5494 | 5570 | 5646 | 5722 | 5799 

570 5875 | 5951 | 6027 | 6103 | 6180 || 6256 6382 | 6408 | 6484 | 6560 
if 6636 | 6712 | 6788 | 6864 | 6940 || 7016 7092 | 7168 | 7244 | 7320 %6 
2 7396 | 747 7548 | 7624 | 7700 || 7775 | 7851 | 7927 | 8003 | 8079 
3 8155 | 8280 | 8306 | 8382 | 8458 || 8533 8609 | 8685 | 8761 | 8836 

e. 4 8912 | 8988 | 9068 | 9139 | 9214 |; 9290 9866 | 9441 | 9517 | 9592 
5 9668 | 9743 | 9819 | 9894 | 9970 a 

—| ——_ |—_—_ |—___ 0045 0121 | 0196 | 0272 | 0347 
6 | 760422 | 0498 | 0573 | 0649 | 0724 || 0799 . 0875 | 0950 | 1025 | 1101 
fi 1176 | J251 | 1826 | 1402 | 1477 1552 | 1627 | 1702 | 1778 | 1853 
8 1928 | 20038 | 2078 | 2153 | 2228 || 2308 | 2378 | 2458 | 2529 | 2604 Ws 
9 267: 71d4 | 2829 | 2904 | 2978 || 8053 |, 3128 | 8208 | 327 8353 

580 8428 | 3503 | 3578 | 3653 | 8727 || 3802 | 38877 | 3952 | 4027 | 4101 
1 4176 | 4251 | 4826 | 4400 | 4475 || 4550 | 4624 | 4699 | 4774 | 4848 
2 4923 | 4998 | 5072 | 5147 | 5221 5296 | 5370 | 5445 | 5520 | 5594 
3 5669 | 5743 | 5818 | 5892 | 5966 || 6041 | 6115 | 6190 | 6264 | 63388 
4 6413 | 6487 | 6562 | 6636 | 6710 || 6785 | 6859 | 6988 | 7007 | 082 

PROPORTIONAL PARTS, 

Diff. 1 2 3 4 5 6 @ 8 9 
83 8.3 16.6 24.9 33.2 41.5 49.8 58.1 66.4 74.9 
82 8.2 16.4 24.6 82.8 41.0 49.2 57.4 65.6 73.8 
81 8.1 16.2 24.3 382.4 40.5 48 .6 5627 64.8 72.9 
80 8.0! 16.0 | 24.0 82.0 40.0 48.0 56.0 64.0 72.0 
79 7.9 15:38 ah lif 81.6 89.5 47.4 55.3 63.2 Tele | 
%8 fhe 15.6 23.4 3132 39.0 46.8 54.6 62.4 70.2 
cue Gok 15.4 23.1 30.8 88.5 46.2 53.9 61.6 69.3 
%6 7.6 15.2 22.8 80.4 388.0 45.6 53.2 |. 60.8 68.4 
%5 7.5 15.0 22.0 30.0 87.5 45.0 52.56 60.0 67.5 
"4 7.4 | 14.8 22.2 29.6 87.0 44.4 51.8 59.2 66 6 


80 TABLE XXIV.—LOGARITHMS OF NUMBERS 


— - —OOooornwn oon Se 


























































































































No. 585 L. 767.] [No. 629 L. 799. 
N 0 1 2 8 4 5 6 7 8 9 | Diff. 
585 | 767156 | 7230 | 73804 | 7379 | 7453 || 7527 | 7601 | 7675 | 7749 | 7823 
6 7898 | 7972 | 8046 | 8120 | 8194 || 8268 | 8342 | 8416 | 8490 | 8564 74 
7 8638 | 8712 | 8786 | 8860 | 8934 || 9008 | 9082 | 9156 | 9230 | 9308 
8 9377 | 9451 | 9525 | 9599 | 9673 || 9746 | 9820 | 9894 | 9968 |——— 
eee | — —— | |__| —__| 0042 
9 | 770115 | 0189 | 0268 | 0386 | 0410 || 0484 | 0557 | 0631 | O705 | 0778 
590 0852 | 0926 | 0999 | 1073 | 1146 || 1220 | 1298 | 1867 ) 1440 | 1514 
if 1587 | 1661 | 17384 | 1808 | 1881 || 1955 | 2028 | 2102 | 2175 | 2248 |. 
2 2322 | 2395 | 2468 | 2542 | 2615 || 2688 | 2762 | 28385 | 2908 | 2981 
3 8055 | 3128 | 8201 | 38274 | 3348 || 8421 | 3494 | 3567 | 3640 | 3713 
4 8786 | 386C | 39383 | 4006 | 4079 || 4152 | 4225 | 4298 | 4371 | 4444 73 
5 4517 | 4590 | 4663 | 4736 | 4809 || 4882 | 4955 | 5028 | 5100 | 5173 
6 5246 | 5319 | 53892 | 5465 | 5588 || 5610 | 5683 | 5756 | 5829 | 5902 
q 5974 | 6047 | 6120 | 6193 | 6265 || 6338 | 6411 | 6483 | 6556 | 6629 
8 6701 | 6774 | 6846 | 6919 | 6992 || 7064 | 7137 | 7209 | 7282 | 7354 
9 7427 | 7499 | 7572 | 7644 | 7717 || 7789 | 7862 | 7934 | 8006 | 8079 
690 8151 | 8224 | 8296 | 8368 | 8441 || 8518 | 8585 | 8658 | 8730 | 8802 
1 8874 | 8947 | 9019 | 9091 | 9163 || 9286 | 9308 | 9380 | 9452 | 9524 
2 9596 | 9669 | 9741 | 9813 | 9885 || 9957 |——— eee 
ee ee eee 0029 | 0101 | 0173 ; 0245 "9 
8 | 780317 | 03889 | 0461 | 0583 | 0605 || 0677 | 0749 | 0821 | 0893 | 0965 
4 1037 | 1109 | 1181 | 1253 | 1324 1396 | 1468 | 1540 | 1612 | 1684 
5 1755 | 1827 | 1899 | 1971 |. 2042 || 2114 | 2186 | 2258 | 2329 | 2401 
6 2473 | 2544 | 2616 | 2688 | 2759 || 2831 | 2902 | 2974 | 3046 | 3117 
q 3189 | 3260 | 3832 | 3403 | 3475 || 3546 | 3618 | 3689 | 8761 | 3832 
8 3904 | 3975 | 4046 | 4118 | 4189 || 4261 | 4332 | 4403 | 4475 | 4546 : 
9 4617 | 4689 | 4760 | 4881 | 4902 || 4974 | 5045 | 5116 | 5187 | 5259 
610 5330 | 5401 | 5472 | 5543 | 5615 || 5686 | 5757 | 5828 | 5899 | 597 
1 6041 | 6112 | 6183 | 6254 | 6825 || 6396 | 6467 | 6538 | 6609 | 6680 71 
2 6751 | 6822 | 68938 | 6964 | 7035 || 7106 | 7177 | 7248 | 7319 | 7390 
3 7460 | 75381 | 7602 | 7673 | 77 7815 | 7885 | 7956 | 8027 | 8098 
4 8168 | 8239 | 8310 | 8881 | 8451 || 8522 | 8593 | 8663 | 8784 | 8804 
5 8875 | 8946 | 9016 | 9087 | 9157 || 9228 | 9299 | 93869 | 9440 | 9510 
6 9581 | 9651 | 9722 | 9792 | 9863 || 9938 aoa 
——— | —_ | ——_ | ——__ | —_ ——-| 0004 | 0074 | 0144 | 0215 
7% | 790285 | 0356 | 0426 | 0496 | 0567 || 0637 | 0707 | O778 | 0848 | 0918 
8 0988 | 1059 | 1129 | 1199 | 1269 || 1840 | 1410 | 1480 | 1550 | 1620 
9 1691 | 1761 | 1831 | 1901 | 1971 2041 | 2111 | 2181 | 2252 | 2329 
620 2392 | 2462 | 2582 | 2602 | 2672 || 2742 | 2812 | 2882 | 2952 | 3022 | 70 
ia 8092 | 3162 | 3231 | 3301 | 3371 || 3441 | 3511 | 3581 | 3651 | 3721 
2 3790 | 3860 | 39380 | 4000 | 407 4139 | 4209 | 4279 | 4349 | 4418 
o 4488 | 4558 | 4627 | 4697 | 4767 || 4836 | 4906 | 4976 | 5045 | 5115 
4 5185 | 5254 | 53824 | 5893 | 5463 || 55382 | 5602 | 5672 | 5741 | 5811 
5 5880 | 5949 | 6019 | 6088 | 6158 || 6227 | 6297 | 6366 | 6436 | 6505 
6 6574 | 6644 | 6713 | 6782 | 6852 || 6921 | 6990 | 7060 | 7129 | 7198 
7 7268 | 738387 | 7406 | 7475 | 7545 || 7614 | 7683 | 7752 | 7821 | 7890 
8 7960 | 8029 | 8098 | 8167 | 8286 || 8305 | 8374 | 8443 | 8513 | 8582 
9 8651 | 8720 | 8789 | 8858 | 8927 || 8906 | 9065 | 9184 | 9203 | 9272 69 
PROPORTIONAL PARTS. 

Diff 1 2 3 4 a 6 G 8 9 
45 ey) 15 E 0, 22.5 30.0 37.5 45.0 52.5 60.0 67.5 
4 Tale 148 Dose 29.6 37.0 44.4 51.8 59.2 66.6 
%3 (uacr 114.6 21.9 29.2 36.5 43.8 51.1 58.4 65.7 
72 Wie Ale 21.6 28.8 36.0 43.2 50.4 57.6 64.8 
val le) del ae Pailaes’ 28.4 35.5 42.6 49.7 56.8 63.9 
%0 7.0 | 14.0 21.0 28.0 85.0 42.0 49.0 56.0 63.0 
69 6.9 | 138.8 20.7 27.6 84.5 41.4 48.3 55.2 62.1 





- 


TABLE XXIV.—LOGARITHMS OF NUMBERS 


81 





No. 630 L. 799.] 




















n.{ 0 (ars aire | Als Cee bk 
630 | 799341 | 9409 | 9478 | 9547 | 9616 || 9685 | 9754 | 9823 | 9892 
1 | 800029 | 0098 | 0167 | 0236 | 0305 || 0873 | 0442 | 0511 | 0580 
Ve 0717 | 0786 | 0854 | 0923 | 0992 || 1061 | 1129 | 1198 | 1266 
3 1494 | 1472 | 1541 |-1609 | 1678 || 1747 | 1815 | 1884 | 1952 
4 2089 | 2158 | 2226 | 2295 | 2363 || 24382 | 2500 | 2568 | 2637 
5 277 2842 | 2910 | 2979 | 38047 || 83116 | 3184 | 3252 | 3321 
6 3457 | 3525 | 3594 | 3662 | 3730 || 3798 | 3867 | 3935 | 4003 
ia 4139 | 4208 76 | 4844 | 4412 || 4480 | 4548 | 4616 | 4685 
8 4821 | 4889 | 4957 | 5025 | 5093 || 5161 | 5229 | 5297 | 5365 
9 5501 | 5569 | 5637 | 5705 | 5778 || 5841 | 5908 | 5976 | 6044 
640 | 806180 | 6248 | 6316 | 63884 | 6451 || 6519 | 6587 | 6655 | 6723 
1 6858 | 6926 | 6994 | 7061 | 7129 || 7197 | 7264 | 7382 | 7400 
2 7535 | 7603 | 7670 | 77388 | 7806 || 7873 | 7941 | 8008 | 8076 
3 8211 | 8279 | 8346 | 8414 | 8481 || 8549 | 8616 | 8684 | 8751 
4 8886 | 8953 | 9021 | 9088 | 9156 || 9223 | 9290 | 9358 | 9425 
5 9560 | 9627 | 9694 | 9762 | 9829 || 9896 | 9964 |———|——-— 
——|-~—__—_| -____] ______||______|______| 0031 | 0098 
6 | 810233 | 0300 | 0367 | 0484 | 0501 0569 | 0636 | 0703 | 0770 
v4 0904 | 0971 | 1039 | 1106 | 1178 || 1240 | 1307 | 137 1441 
8 1575 | 1642 | 1709 | 1776 | 1848 1910 | 1977 | 2044 | 2111 
9 2945 | 2812 | 23879 | 2445 | 2512 || 2579 | 2646 | 2718 | 2780 
650 2913 | 2980 | 3047 | 3114 | 3181 || 3247.) 3314 | 3381 | 3448 
1 3581 | 3648 | 3714 | 3781 | 3848 || 3914 | 3981 | 4048 | 4114 
2 4248 | 4814 | 4881 | 4447 | 4514 || 4581 | 4647 | 4714 | 4780 
3 4913 | 4980 | 5046 | 5118 | 5179 || 5246 | 5312 | 5378 | 5445 
4 5578 | 5644 | 5711 | 5777 | 5848 || 5910 | 597 6042 | 6109 
5 6241 | 6308 | 6374 | 6440 | 6506 || 657 6639 | 6705 | 6771 
6 6904 | 697 7036 | 7102 | 7169 || 7235 | 73801 | 73867 | 7433 
fi 7565 | 76381 | 7698 | 7764 | 7830 || 7896 | 7962 | 8028 | 8094 
8 8226 | 8292 | 8858 | 8424 | 8490 || 8556 | 8622 | 8688 | 8754 
ae!) 8885 | 8951 9017 9083 | 9149 ||} 9215 | 9281 | 9346 | 9412 
660 9544 | 9610 | 9676 | 9741 | 9807 || 9873 | 9939 | 
3 ———_ |—_——_ —_—_ | | —___ 0004 | 0070 
1 | 820201 | 0267 | 0333 | 0399 | 0464 || 0530 | 0595 | 0661 | 0727 
2 0858 | 0924 | 0989 | 1055 | 1120 || 1186 | 1251 | 1317 | 1882 
3 $514 | 157 1645 |.1710 | 177 1841 | 1906 | 1972 | 2087 
4 2168 | 2233 | 2299 | 2364 | 2480 || 2495 | 2560 | 2626 | 2691 
5 2822 | 2887 | 2952 | 3018 | 3083 || 3148 | 3213 | 38279 | 3844 
6 3474 | 3539 | 3605 | 3670 | 3735 || 3800 | 8865 | 39380 | 3996 
ie 4126 | 4191 | 4256 | 4821 | 4386 || 4451 | 4516 | 4581 | 4646 
8 4776 | 4841 | 4906 | 4971 | 5036 || 5101 | 5166 | 5281 | 5296 
9 5426 | 5491 | 5556 | 5621 | 5686 || 5751 | 5815 | 5880 | 5945 
670 6075 | 6140 | 6204 | 6269 | 6334 || 6899 | 6464 | 6528 | 6593 
1 6723 | 6787 | 6852 | 6917 | 6981 || '7046 | 7111 | 7175 | 7240 
2 369 | 7434 | 7499 | 7563 | '7628 || 7692 | 7757 | 782 7886 
3 8015 | 80890 | 8144 | 8209 | 827 8338 | 8402 | 8467 | 8531 
4 8660 | 8724 | 8789 | 8853 | 8918 || 8982 | 9046 | 9111 | 9175 
| 
PROPORTIONAL PARTS, 
Diff. i! a 3 4 5 6 i 
68 6.8 13.6 20.4 27.2 34.0 40.8 47.6 
i 6.7 | 13.4 20.1 26.8 33.5 40.2 46.9 
66 6.6 13.2 19.8 26.4 33.0 39.6 46.2 
65 6.5 13.0 19.5 26.0 32.5 39.0 45.5 
64 6.4 12.8 19.2 25.6 32.0 88.4 44.8 
















































































[No. 674 L. 829. 


9 | Diff. 


67 








66 





65 





8 9 


61.2 
60.3 
59.4 
58.5 
57.6 


54.4 
53.6 
52.8 
52.0 
51.2 





82 TABLE XXIV.—LOGARITHMS OF NUMBERS 





No. 675 L. 829.] No. 719 L. 857. 





N. 0 1 2 3 4 5 6 7 8 9 | Diff. 


675 | 8293804 | 9368 | 9432 | 9497 | 9561 || 9625 | 9690 | 9754 | 9818 | 9882 
6 9947 





830589 | 0653 | O717 | O781 | 0845 || 0909 | 0973 | 1037 | 1102 | 1166 














8219 | 9282 | 8345 | 8408 | 8471 || 9534 | 8597 | 8660 | 8723 | 8786 | %8 














5718 | 5780 | 5842 | 5904 | 5966 || 6028 | 6090 | 6151 | 6213 | 6275 62 








2480 | 2541 | 2602 | 2663 | 2724 || 2785 | 2846 | 2907 | 2968 | 3029 | SL 








0 

i 

2 

3 

4 

5 

6 

fy 

8 : 
9 0646 | 0707 | 0769 | 0880 | O891 || 0952 | 1014 | 1075 | 1136 | 1197 
0 

1 

2 

3 

4 

5 

6 

7 

8 

9 








pift.| 1 | 2 3 4 5 6 7 8 9 
65 | 6.5 | 13.0 -| 19.5 | 26.0 | 32.51 29.0.4 45 + 52.0 | 58:5 
64 | 64 | 12.8 | 19.2.1 25.6 | 32.0.1 B84] 44.80) Be | $7.6 
63 | 631 12.6 | 18.9 | 25.2 1 31.5:| 37.8 |] 44.1 | 50.4 | 56.7 
eo | 62) 12.4 | 18.6 | 24.8 | 31.0 | 37.2 | 43.4 | 49.6 | 55.8 
61 | 6.1 12.2 | 18.3 | 24.4-| 30.5 | 36.6 | 42.7 | 48.8 | 54.9 
eo | 6.0 | 12.0 ; 18.0 | 24.0 | 30.0 | 36.0 | 42.0 | 48.0 | 54-0 


TABLE XXIV.—LOGARITHMS OF NUMBERS 83 





































































































No. 720 L. 857:] ; [No. 764 L. 888. 
N.| 0 1 2 3 4 5 6 "y 8 9 eee, 
720 | 857332 | 7393 | 7453 | 7513 | 7574 || 7634 | 7694 | 7755 | 7815 | 79875 
1| 7935 | 7995 | 8056 | 8116 | 8176 || 8236 | 8297 | 8357 | 8417 | 8477 
2 | 9537 | 9597 | 8657 | 8718 | 8778 || 8838 | 8898 | 8958 | 9018 | 9078 
3 9138 | 9198 | 9258 | 9318 | 9379 || 9439 | 9499 | 9559 | 9619 | 9679 | 60 
4 | 9739 {9799 | 9859 |'9918,| 9978 || ———|—__|__|—____|. __ 
_—_—_| —_|___|___]| 0038 | 0098 | 0158 | 0218 | 0278 
5 | 860338 | 0398 | 0458 | 0518 | 0578 || 0637 | 0697 | 0757 | O817 | 0877 
6 | 0937 | 0996 | 1056 | 1116 | 1176 || 1236 | 1295 | 1355 | 1415 | 1475 
7 | 4534 | 1594 | 1654 | 1714 | 1773 || 1833 | 1893 | 1952 | 2012 | 2072 
8 | 2131 | 2191 | 2251 | 2310 | 2370 |] 2430 | 2489 | 2549 | 2608 | 2668 
9 | 2728 | 2787 | 2847 | 2906 | 2966 || 3025 | 3085 | 3114 | 3204 | 3263 
730 | 3823 | 3382 | 3442 | 3501 | 3561 || 3620 | 3680 | 3739 | 3799 | 3858 
1| 3917 | 3977 | 4036 | 4096 | 4155 || 4214 | 4274 | 4333 | 4392 | 4452 
2| 4511 | 4570 | 4630 | 4689 | 4748 || 4808 | 4867 | 4926 | 4985 | 5045 
3 | 5104 | 5163 | 5222 | 5282 | 5341 || 5400 | 5459 | 5519 | 5578 | 5637 
4| 5696-| 5755 | 5814 | 5874 | 5933 || 5992 | 6051 | 6110 | 6169 | 6228 
5 | 6287 | 6346 | 6405 | 6465 | 6524 || 6583 | 6642 | 6701 | 6760 | 6819 | 5g 
6 | 6878 ; 6937 | 6996 | 7055 | 7114 || 7173 | 7232 | 7291 | 7350 | 7409 
7 | 7467 | 7526 | 7585 | 7644 | 7703 || 7762 | 7821 | 7880 | 7939 | 7998 
8 | 8056 | 8115 | 8174 | 8233 | 8292 || 8350 | 8409 | 8468 | 8527 | 8586 
9 | 98644 | 8703 | 8762 | $821 | 8879 || 8938 | 8997 | 9056 | 9114] 9173 
740 | 9282 | 9290 | 9349 | 9408 | 9466 || 9525 | 9584 | 9642 | 9701 | 9760 
1| 9818 | 9877 | 9935 | 9994 ||| ___ ees ai Ed 
-_——| 0053 || 0111 | 0170 | 0228 | 0287 | 0345 
2 | 870404 | 0462 | 0521 | 0579 | 0638 || 0696 | 0755 | 0813 | 0872 | 0930 
3 | 0989 | 1047 | 1106 | 1164 | 1223 || 1981 | 1339 | 1398 | 1456 | 1515 
4 1573 | 1631 | 1690 | 1748 | 1806 || 1865 | 1923 | 1981 | 2040 | 2008 
5 | 2156 | 2215 | 2273 | 2331 | 2389 |) 2448 | 2506 | 2564 | 2622 | 2681 
6 | 2739 | 2797 | 2855 | 2913 | 2972 || 3030 | 3088 | 3146 | 3204 | 3262 
7” | 3307 | 3379 | 3437 | 3495 | 3553 |) 3611 | 3669 | 3727 | 3785 | 3844 
8 | 3902 | 3960 | 4018 | 4076 | 4134 || 4192 | 4250 | 4308 | 4366 | 4424 | 58 
- 9 | 4482 | 4540 | 4598 | 4656 | 4714 || 4772 | 4830 | 4888 | 4945 | 5003 
750 | 5061 | 5119 | 5177 | 5285 | 5293 || 5351 | 5409 | 5466 | 5524 | 5582 
1| 5640 | 5698 | 5756 | 5813 | 5871 || 5929 | 5987 | 6045 | 6102 | 6160 
2| 6218 | 6276 | 6333 | 6391 | 6449 || 6507 | 6564 | 6622 | 6680 | 6737 
3 | 6795 | 6853 | 6910 | 6968 | 7026 || 7083 | 7141 | 7199.| 7256 | 7314 
4| 7371 | 7429 | 7487 | 7544 | z602 || 7659 | 7717 | 7774 | 7932 | 7889 
5'| 947 | 8004 | 8062 | 8119 | 8177 || 8234 | 8292 | 8849 | 8407 | 8464 
6 | 8522 | 8579 | 8637 | 8694 | 8752 || 8809 | 8866 | 8924 | 8981 | 9039 
7 | 9096 | 9153 | 9211 | -9268 | 9325 |) 9383 | 9440 | 9497 | 9555 | 9612 
8 | 9669 | 9726 | 9784 | 9841 | 9898 || 9956 |——— Pde bes 7g 
yee e -||——] 0013 | 0070 | 0127 | 0185 
9 | 880242 | 0299 | 0356 | 0413 | 0471 || 0528 | 0585 | 0642 | 0699 | 0756 
760 | 0814 | 0871 | 0928 | 0985 | 1042 || 1099 | 1156 | 1213 | 1271 | 1328 
1| 1385 | 1442 | 1499 | 1556 | 1613 || 1670 | 1727 | 1784 | 1841 | 1898 
2| 1955 | 2012 | 2069 | 2126 | 2183 || 2240 | 2207 | 2354 | 2411 | 2468] 5? 
3 | 2525 | 2581 | 2638 | 2695 | 2752 || 2809 | 2866 | 2923 | 2980 | 3037 
4| 3093 | 3150 | 3207 | 3264 | 3321 || 3377 | 3434 | 3491 | 3548 | 3605 
PROPORTIONAL PARTS. 
Difft.| 1 2 3 4 5 6 " 8 9 


84 TABLE XXIV.—LOGARITHMS OF NUMBERS 





"No. 765 L. 883.] [No. 809 L. 908, 















































765 | 883661 | 3718 | 3775 | 3832 | 3888 || 3945 | 4002 | 4059 | 4115 | 4172 
6 4229 | 4285 | 4842 | 4399 | 4455 || 4512 | 4569 | 4625 | 4682 | 4739 
7 4795 | 4852 | 4909 | 4965 | 5022 || 5078 | 5185 | 5192 | 5248 | 53805 
8 5861 | 5418 | 5474 | 5531 | 5587 || 5644 | 5700 | 5757 | 58138 | 5870 
9 5926 | 5983 | 6039 | 6096 | 6152 || 6209 | 6265 | 6821 | 6387 
770 6491 | 6547 | 6604 | 6660 | 6716 || 6773 | 6829 | 6885 | 6942 | 6998 
ie 7054 | 7111 | 7167 | 7223 | 7280 || 7336 | 7392 | 7449 | 7505 | 7561 
2 7617 | 7674 | 7730 | 7786 | 7842 || 7898 | 7955 | 8011 | 8067 | 8123 | 
3 8179 | 8236 | 8292 | 8348 | 8404 || 8460 | 8516 | 8573 | 8629 | 8685 
4 8741 | 8797 | 8853 | 8909 | 8965 || 9021 | 9077 | 9134 | 9190 | 9246 
5 9302 | 9358 | 9414 | 9470 | 9526 || 9582 | 9638 | 9694 | 9750 | 9806 56 
6 9862 | 9918 | 9974 —_—— oa 
——| 0030 | 0086 || 0141 | 0197 | 0253 | 0309 | 0365 
7 | 890421 | 0477 | 0533 | 0589 | 0645 || 0700 | 0756 | 0812 | 0868 | 0924 
8 0980 | 1035 | 1091 | 1147 | 1203 || 1259 | 1314 | 1370 | 1426 | 1482 
9 1537 | 1593 | 1649 | 1705 | 1760 || 1816 | 1872 | 1928 | 1983 | 2039 
780 2095 | 2150 | 2206 | 2262 | 2317 || 2373 | 2429 | 2484 |. 2540 | 2595 
1 2651 | 2707 | 2762 | 2818 | 287 2929 | 2985 | 3040 | 3096 | 3151 
2 8207 | 8262 } 3318 | 3373 | 3429 || 3484 | 3540 | 3595 | 3651 | 3706 
3 3762 | 3817 | 3873 | 3928 | 3984 || 4039 | 4094 | 4150 | 4205 | 4261 
4 4316 | 4371 | 4427 | 4482 | 4538 || 4593 | 4648 | 4704 | 4759 | 4814 
5 4870 | 4925 | 4980 | 5036 | 5091 || 5146 | 5201 | 5257 | 5312 | 5867 
6 5423 | 5478 | 5533 | 5588 | 5644 || 5699 | 5754 | 5809 | 5864 | 5920 
7 5975 | 6030 | 6085 | 6140 | 6195 || 6251 | 6306 | 6361 | 6416 | 647 
8 6526 | 6581 | 6636 | 6692 | 6747 || 6802 | 6857 | 6912 | 6967 | 7022 
9 VOT? | 7132 | 7187 | 7242 | 7297 || 73852 | 7407 | 7462 | 7517 | 7572 BB 
790 7627 | 7682 | 7737 | 7792 | 7847 || 7902 | 7957 | 8012 | 8067 | 8122 ‘. 
1 8176 | 8231 | 8286 | 8341 | 83896 || 8451 | 8506 | 8561 | 8615 | 8670 
2 87 8780 | 8835 | 8890 | 8944 || 8999 | 9054 | 9109 | 9164 | 9218 
3 9273 } 9828 | 9383 | 9437 | 9492 || 9547 | 9602 | 9656 | 9711 | 9766 
4 9821 | 9875 | 9930 | $985 Sea tS eel Se eae 
a | I 0089 1] 0094 | 201492 10208 120258: Ee es2 
5 | 9003867 | 0422 | 0476 | 0531 | 0586 || 0640 | 0695 | 0749 | 0804 | 0859 
6 0913 | 0968 | 1022 | 1077 | 1181 || 1186 | 1240 | 1295 | 1349 | 1404 
7 1458 | 1513 | 1567 | 1622 | 1676 || 1731 | 1785 | 1840 | 1894 | 1948 
8 2003 | 2057 | 2112 | 2166 | 2221 || 2275 | 2329 | 23884 | 2438 | 2492 
9 2547 | 2601 | 2655 | 2710 | 2764 || 2818 | 2878 | 2927 | 2981 | 3036 
800 3090 | 3144 | 3199 | 3253 | 3307 || 3361 | 3416 | 3470 | 3524 | 3578 
1 3633 | 3687 | 3741 | 3795 | 3849 || 3904 | 3958 | 4012 | 4066 | 4120 
2 4174 | 4229 | 4283 | 4337 | 4391 || 4445 | 4499 | 4553 | 4607 | 4661 
3 4716 | 4770 | 4824 | 4878 | 4982 || 4986 | 5040 | 5094 | 5148 | 5202 
4 5256 | 5310 | 5364 | 5418 | 5472 || 5526 | 5580 | 5634 | 5688 | 5742 | 54 
5 5796 | 5850 | 5904 | 5958 | 6012 || 6066 | 6119 | 6173 | 6227 | 6281 
6 6335 | 6389 | 6443 | 6497 | 6551 || 6604.| 6658 | 6712 | 6766 | 6820 
7 6874 | 6927 | 6981 | 7035 | 7089 || 7143 | 7196 | 7250 | 7304 | 7358 
8 7411 | 7465 | 7519 | 7573 | 7626 || 7680 | 77384 | 7787 | 7841 | 7895 
9 7949 | 8002 | 8056 | 8110 | 8163 || 8217 | 8270 | 8324 | 8378 | 8431 
a 
PROPORTIONAL PARTS. 
Diff. 1 2 3 4 5 6 7 8 9 
57 11.4 Ye 22.8 28.5 


TABLE XXIV.—LOGARITHMS OF NUMBERS ~~ 85 


————— eet 











































































































No. 810 L, 908.] [No. 854 L. 931. 
IN: 0 1 2 3 4 5 6 7 8 Om | Dit: 
ee eae eee | Ree Se ee 
* 

810 | 908485 | 8539 | 8592 | 8646 | 8699 || 8753 | 8807 | 8860 | 8914 | 8967 
1 9021 | 9074 | 9128 | 9181 | 9235 || 9289 | 9342 | 9396 | 9449 | 9508 
2 9556 | 9610 | 9663 | 9716 | 977 9823 | 9877 | 9930 | 9984 |——— 

—. ll ——— 0037 
3 | 910091 | 0144 | 0197 | 0251 | 0304 0358 | 0411 | 0464 | 0518 | 0571 
4 0624 | 0678 | 0731 | 07 0838 |} 0891 | 0944 | 0998 | 1051 | 1104 
5 1159) eA E64 ely 4) Sa 1424 | 147 1530 | 1584 | 1637 
6 1690 | 1743 | 1797 | 1850 | 1908 1956 | 2009 | 2063 | 2116 | 2169 
ii 2222") 220 2328 | 2881 | 2435 2488 | 2541 | 2594 | 2647 | 2700 
8 753 | 2806 | 2859 | 2913 | 2966 || 3019 | 3072 | 3125 | 8178 | 3231 
9 3284 | 3337 | 3390 | 3443 | 3496 || 8549 | 38602 | 8655 | 3708 | 3761 53 

820 3814 | 38867 | 3920 | 3973 | 4026 || 4079 | 4132 | 4184 | 4287 | 4290 
1 4343 | 4396 | 4449 | 4502 | 4555 |! 4608 | 4660 | 4718 ! 4766 | 4819 
Q 4872 | 4925 | 4977 | 5030 | 5083 || 5136 | 5189 | 5241 | 5294 | 5347 
3 5400 | 5453 | 5505 | 5558 | 5611 || 5664 | 5716 | 5769 | 5822 | 5875 
4 5927 | 5980 | 6033 | 6085 | 6138 || 6191 | 6243 | 6296 | 6349 | 6401 
5 6454 | 6507 | 6559 | 6612 | 6664 || 6717 | 6770 | 6822 | 6875 | 6927 
6 6980. |} 7033! 7085 | 7188 | 7190 || 7248 | 7295 | 7348 | 7400 | 7453 
vi 7506 | 7558 | 7611 | 7663 | 7716 || 7768 | 7820 | 7873 | 7925 | 7978 
8 8030 | 8083 | 8135 | 8188 | 8240 || 8293 | 8345 | 8397 | 8450 | 8502 
9 8555 | 8607 | 8659 | 8712 | 8764 || 8816 | 8869 | 8921 | 8973 | 9026 

830 9078 | 91380 | 9183 | 9235 | 9287 || 9340 | 9392 | 9444 | 9496 | 9549 
1 9601 | 9658 | 9706 | 9758 | 9810 || 9862 | 9914 | 9967 |———-|__—_ 

AN aes Se eet eet eee (eee oe 0019 | 0071 
2 | 9201238 | 0176 | 0228 | 0280 | 0332 || 0884 | 0486 | 0489 | 0541 | 0593 
3 0645 | 0697 | 0749 | 0801 | 0853 0906 | 0958 | 1010 | 1062 | 1114 58 
4 1166 | 1218 | 1270 | 1822 | 137 1426 | 1478 | 1530 | 1582 | 1634 See 
5 1686 | 1738 | 1790 | 1842 | 1894 || 1946 | 1998 | 2050 | 2102) 2154 
6 2206 | 2258 | 2310 | 2362 | 2414 || 2466 | 2518 | 2570 | 2622 | 2674 
7 720 | 2777 | 2829 | 2881 | 2933 || 2985 | 3037 | 3089 | 3140 | 3192 

nS 3244 | 8296 | 3348 | 3399 | 3451 || 3503 | 3555 | 38607 | 8658 | 3710 
9 762 | 3814 | 3865 | 3917 | 3969 || 4021 | 4072 | 4124 | 4176 | 4228 

840 4279 | 4331 | 4383 | 4484 | 4486 || 4538 | 4589 | 4641 | 4698 | 4744 
il 4796 | 4 4899 | 4951 | 5008 || 5054 | 5106 | 5157 | 5209 | 5261 
2 53812 | 53864 | 5415 | 5467 | 5518 55 5621 | 567 725 | 5776 
3 5828 | 5879 | 5931 | 5982 | 6034 || 6085 | 6187 | 6188 | 6240 | 6291 
4 6342 | 6394 | 6445 | 6497 | 6548 6600 | 6651 | 6702 | 6754 | 6805 
5 6857 | 6908 | 6959 | 7011 | 7062 || 7114 | 7165 | 7216 | 7268 | 7319 
6 7370 | 7422 | 7473 | 7524 | 7576 || 7627 | 7678 | 7730 | 7781 | 78382 
7 7883 | 7935 | 7986 | 8037 | 8088 || 8140 | 8191 | 8242 | 8298 | 8345 
8 8396 | 8447 | 8498 | 8549 | 8601 || 8652 | 8703 | 8754 | 8805 | 8857 
9 8908 | 8959 | 9010 | 9061 | 9112 || 9163 | 9215 | 9266 | 93817 | 9868 

850 9419 | 9470 | 9521 | 9572 | 9623 || 9674 | 9725 | 9776 | 9827 | 9879 Bt 
1 99507 1) 008). | | es 

—— 0032 | 0083 | 0134 |) 0185 | 0236 | 0287 | 03388 | 0389 

2 | 930440 } 0491 | 0542 | 0592 | 0648 || 0694 | 0745 | 0796 | 0847 | 0898 

3 0949 | 1000 | 1051 | 1102 | 1153 1204 | 1254 | 1805 | 13856 | 1407 

4 1458 | 1509 | 1560 | 1610 | 1661 1712 | 1763 | 1814 | 1865 | 1915 
PROPORTIONAL PARTS. 

Diff. 1 | 2 see) 4 5 6 if 8 9 
53 Bonin 106 15.9 PIE. 26.5 31.8,|- 37.1 42.4 47.7 
52 5.2) 10.4 15.6 20.8 26.0 31.2 86.4 41.6 46.8 
51 5.1 10.2 1513 20.4 25.5 30.6 85.7 40.8 45.9 
50 5.0 | 10.0 15.0 20.0 25.0 80.0 35.0 40.0 45.0 








86 TABLE XXIV.—LOGARITHMS OF NUMBERS 

















































































































No. 855 L, 931.] LNo. 899 Li. 954, 

N. 0 1 2 3 4 5 6 7 8 9 | Diff. 

855 | 931966 | 2017 | 2068 | 2118 | 2169 || 2220 | 2271 | 2822 | 2372 | 2423 
6 2474 | 2524 | 2575 | 2626 | 2677 || 2727 | 2778 | 2829 | 2879 | 2930 
tf 2981 | 3081 | 8082 | 381383 | 3188 || 3234 | 8285 | 33835 | 3386 | 3437 
8 8487 | 3538 | 8589 | 3639 | 8690 || 38740 | 3791 | 38841 | 8892 | 3943 
9 8998 | 4044 | 4094 | 4145 | 4195 || 4246 | 4296 | 4347 | 4897 | 4448 

860 4498 | 4549 | 4599 | 4650 | 4700 || 4751 | 4801 | 4852 | 4902 | 4953 
1 5003 | 5054 | 5104 | 5154 | 5205 || 5255 | 5306 | 5356 | 5406 | 5457 
2 5507 | 5558 | 5608 | 5658 | 5709 || 5759 | 5809 | 5860 | 5910 | 5960 
3 6011 | 6061 | 6111 | 6162 | 6212 || 6262 | 6313 | 6363 | 6413 | 6463 
4 6514 | 6564 | 6614 | 6665 | 6715 || 6765 | 6815 | 6865 | 6916 | 6966 
5 7016 | 066 | 7116 | 7167 | 7217 || 7267 | 7317 | 7367 | '7418 | 7468 
6 7518 | 7568 | 7618 | 7668 | 7718 || 7769 | 7819 | 7869 | 7919 | 7969 50 
% 8019 | 8069 | 8119 | 8169 | 8219 || 8269 | 8820 | 8870 | 8420 | 8470 
8 8520 | 8570 | 8620 | 8670 | 8720 || 8770 | 8820 | 887 8920 | 897 
9 9020 | 9070 | 9120 | 9170 | 9220 || 9270 | 9820 | 9369 | 9419 | 9469 

870 9519 | 9569 | 9619 | 9669 | 9719 || 9769 | 9819 | 9869 | 9918 | 9968 
1 | 940018 | 0068 | 0118 | 0168 | 0218 || 0267 | 03817 | 0867 | 0417 | 0467 
2 0516 | 0566 | 0616 | 0666 | 0716 || 0765 | 0815 | 0865 | 0915 | 0964 
3 1014 | 1064 | 1114 | 1168 | 1218 || 1268 | 1318 | 1862 | 1412 | 1462 
4 1511 | 1561 | 1611 | 1660 | 1710 || 1760 | 1809. | 1859 | 1909 | 1958 
5 2008 | 2058 | 2107 | 2157 | 2207 || 2256 | 2806 | 2855 | 2405 | 2455 
6 2504 | 2554 | 2608 | 2653 702 || 2752 | 2801 | 2851 | 2901 | 2950 
MG 8000 | 3049 | 8099 | 38148 | 3198 || 3247 | 3297 | 3346 | 3396 | 8445 
8 | _ 8495 | 3544 | 3593 | 3643 | 3692 || 3742 | 3791 | 8841 | 8890 | 3939 
9 8989 | 4088 | 4088 | 4187 | 4186 || 4236 | 4285 | 4885 | 4884 | 4433 

880 4483 | 45382 | 4581 | 4631 | 4680 || 4729 | 4779 | 4828 | 4877 | 4927 
1 4976 | 5025 | 5074 | 5124 | 5178 || 5222 | 5272 | 5821 | 5370 | 5419 
2 5469 | 5518 | 5567 | 5616 | 5665 || 5715 | 5764 | 5818 | 5862 | 5912 
3 5961 | 6010 | 6059 | 6108 | 6157 || 6207 | 6256 | 6805 | 6354 | 6403 
4 6452 | 6501 | 6551 | 6600 | 6649 || 6698 | 6747 | 6796 | 6845 | 6894 
5 6943 | 6992 | 7041 | 7090 | 7139 || 7189 | 7238 | 7287 | 7336 | 7385 49 
6 7434 | 7483 | 75382 | 7581 | 76380 || 7679 | 7728 | 7777 | 7826 | 787 
7 7924 | 7973 | 8022 | 8070 | 8119 || 8168 | 8217 | 8266 | 8315 | 8364 
8 8413 | 8462 | 8511 | 8560 | 8608 || 8657 | 8706 755 | 8804 | 8853 
9 8902 | 8951 | 8999 | 9048 | 9097.1] 9146 | 9195 | 9244 | 9292 | 9341 

890 9390 | 9439 | -9488 | 9536 | 9585 || 9634 | 9683 | 9731 | 9780 | 9829 
1 987 9926 | 9975 ——— 

-—— 0024 | 0078 || 0121 | 0170 | 0219 | 0267 | 0316 
2 | 950365 | 0414 | 0462 | 0511 | 0560 || 0608 | 0657 | 0706 | 0754 | 0803 
3 0851 | 0900 | 0949 | 0997 | 1046 |! 1095 | 1143 | 1192 | 1240 | 1289 
4 1338 | 13886 | 1435 | 1488 | 1582 || 1580 | 1629 | 167 a Weta: | on br dga) 
ip 1823 | 1872 | 1920 | 1969 | 2017 || 2066 | 2114 | 2163 | 2211 | 2260 
6 2308 | 2856 | 2405 | 2453 | 2502 || 2550 | 2599 | 2647 | 2696 | 2744 
i 2792 | 2841 | 2889 | 2938 | 2986 || 3034 | 8083 | 3131 | 38180 | 3228 
8 3827 8325 | 38373 | 3421 | 8470 || 3518 | 3566 | 8615 | 3663 | 3711 
9 38760 | 3808 | 3856 | 3905 | 8958 || 4001 | 4049 | 4098 | 4146 | 4194 

PROPORTIONAL PARTS. 

Diff. 1 2 | 3 | 4 5 6 7 8 9 
51 5.1 10.2 15.3 20.4 25.5 30.6 85.7 40.8 45.9 
50 5.0 10.0 15.0 20.0 25.0 80.0 35.0 40.0 45.0 
49 4.9 9.8 14.7 19.6 24.5 29.4 34.3 39.2 44.1 
48 4.8 9.6 14.4 19.2 24.0 28.8 83.6 38.4 43.2 





TABLE XXIV.—LOGARITHMS OF NUMBERS 87 


No 900 L, 954.] [No. 944 L. 975, 




















































































































N. 0 1 2 3 4 5 6 v| 8 9 Diff. 
900 | 954243 | 4291 | 4839 | 4887 | 4485 || 4484 | 4582 | 4580 | 4628 | 4677 
1 4725 | 4773 | 4821 | 4869 | 4918 || 4966 | 5014 | 5062 | 5110 | 5158 
2 5207 | 5255 | 5308 | 53851 | 5899 || 5447 | 5495 | 5543 | 5592 | 5640 
3 5688 | 5736 | 5784 | 58382 | 5880 || 5928 | 5976 | 6024 | 607 6120 
4 6168 | 6216 | 6265 | 6318 | 6361 6409 | 6457 | 6505 | 6553 | 6601 48 
5 6649 | 6697 | 6745 | 6793 | 6840 || 6888 | 6986 | 6984 | 7032 | 7080 
6 7128 | 7176 | 7224 | 7272 | 7320 |) 7368 | 7416 | 7464 | 7512 | 7559 
v4 7607 | 7655 | 7703 | 7751 | 7799 || 7847 | 7894 | 7942 | 7990 | 8038 
8 8086 | 8134 ; 8181 | 8229 | 8e77 || 882 837 8421 | 8468 | 8516 
9 8564 | 8612 | 8659 | 8707 | 8755 || 8803 | 8850 | 8898 | 8946 | 8994 
910 9041 | 9089 | 9187 | 9185 | 9232 || 9280 | 9328 | 9875 | 9423 | 9471 
he 9566 | 9614 | 9661 | 9709 || 9757 | 9804 | 9852 | 9900 | 9947 
eS Es ee | Lee 
— 0042 | 0090 | 0188 | 0185 ||} 02383 | 0280 | 0828 | 03876 | 0423 
3 | 960471 | 0518 | 0566 | 0613 | 0661 || 0709 | 0756 | O804 | 0851 | 0899 
4 0946°| 0994 | 1041 | 1089 | 1136 1184 | 1281 | 1279 | 1326 | 137 
5 1421 | 1469 | 1516 | 1563 | 1611 1658 | 1706 | 1753 | 1801 | 1848 
6 1895 | 1943 | 1990 | 2088 | 2085 || 2182 | 2180 | 2227 | 2275 | 2322 
i 2369 | 2417 | 2464 | 2511 | 2559 || 2606 | 2653 | 2701 | 2748 | 2795 
8 2845 | 2890 | 2937 | 2985 | 3032 || 307 3126 | 3174 | 38221 | 3268 
9 3316 | 33863 | 8410 | 3457 | 38504 || 38552 | 3599 | 3646 | 3698 | 3741 
920 on 3835. | 38882 -| 3929 | 38977 || 4024 | 4071 | 4118 | 4165 | 4212 
1 4260 | 4807 | 4854 | 4401 | 4448 || 4495 | 4542 | 4590 | 4687 | 4684 
2 4731 | 4778 | 4825 | 4872 | 4919 || 4966 | 5013 | 5061 . 5108 | 5155 
3 5202 | 5249 | 5296 | 53438 | 5390 || 5487 | 5484 | 5531 | 5578 | 5625 
4 5672 | 5719 | 5766 | 5813 | 5860 || 5907 | 5954 | 6001 | 6048 | 6095 4f 
13) 6142 | 6189 | 6236 | 6288 | 6829 || 637 6423 | 6470 | 6517 | 6564 
6 6611 | 6658 | 6705 | 6752 | 6799 || 6845 | 6892 | 6989 | 6986 | 7033 
‘a 7080 | 7127 | 7173 | 7220 | 7267 || 7314 | 7861 | 7408 | 7454 | 7501 
8 7548 | 7595 | 7642 | 7688 |.7735 || 7782 | 7829 | 7875 | 7922 | '7969 
ay 8016 | 8062 | 8109 | 8156 | 8203 || 8249 | 8296 | 8343 | 83890 | 8436 
930 8483 | 8530 | 8576 | 8623 | 8670 || 8716 | 8763 | 8810 | 8856 | 8903 
1 8950 | 8996 | 9043 | 9090 | 9136 || 9183 | 9229 | 9276 | 9323 | 9369 
2 9416 | 9463 | 9509 | 9556 | 9602 || 9649 | 9695 | 9742 | 9789 | 9835 
3 9882 | 9928 | 9975 —— —— ——. 
— a 0021 | 0068 || 0114 | 0161 | 0207 | 0254 | 0300 
4 | 970847 | 0393 | 0440 | 0486 | 0533 || 0579 | 0626 | 0672 | 0719 | 0765 
5 0812 | 0858 | 0904 | 0951 | 0997 || 1044 | 1090 | 1187 | 1183 | 1229 
26 1276 | 13822 | 1369 | 1415 | 1461 1508 | 1554 | 1601 | 1647 | 1693 
7 1740 | 1786 | 1832 | 187 1925 1971 | 2018 | 2064 | 2110 | 2157 
8 2203 | 2249 | 2295 | 2342 | 23888 || 2484 | 2481 | 2527 | 2573 | 2619 
9 2666 712 | 2758 | 2804 | 2851 2897 | 2943 | 2989 | 3035 | 3082 
940 | 8128 | 3174 | 3220 | 3266 | 3313 || 8359 | 8405 | 8451 | 8497 | 3543 
1 3590 | 3636 | 3682 | 3728 | 377. 3820 | 3866 | 8913 | 8959 | 4005 
2 4051 | 4097 | 4143 | 4189 | 4235 || 4281 | 4827 | 4874 | 4420 | 4466 
3 4512 | 4558 | 4604 | 4650 | 4696 || 47 4788 | 4834 | 4880 | 4926 
4 4972 | 5018 | 5064 | 5110 | 5156 || 5202 | 5248 | 5294 | 5840 | 5386 46 
PROPORTIONAL PARTS, 
Diff. 1 | te 3 4 | 5 6 ff 8 9 
47 4.7 9.4 14.1 18.8 | 23.5 28.2 S298 1) 37.6 42.3 
46 4.6 9.2 13.8 18.4 23.0 PGA) 82.2 36.8 41.4 








88 TABLE XXIV.—LOGARITHMS OF NUMBERS 





No. 945 L. 975.] [No. 989 L. 995, 




























































































N. 0 1 2 3 4 5 6 fi 8 9 | Diff. 

945 | 975482 | 5478 | 5524 | 5570 | 5616 || 5662 | 5707 | 57538 | 5799 | 5845 
6 5891 | 5987 | 5983 | 6029 | 6075 || 6121 | 6167 | 6212 | 6258 | 6304 
a 6350 | 6896 | 6442 | 6488 | 6583 || 6579 | 6625 | 6671 | 6717 | 6763 
8 6808 | 6854 | 6900 | 6946 | 6992 || 7037 | 7088 | 7129 | 7175 | 7220 
9 | 7266 | 7312 | 7358 | 7403 | 7449 || 7495 | 7541 | 7586 | 7632 | 7678 

950 7724 | 7769 | 7815 | 7861 | 7906 || 7952 | 7998 | 8043 | 8089 | 8135 
1 8181 | 8226 | 8272 | 8317 | 83863 || 8409 | 8454 | 8500 | 8546 | 8591 
2 8637 | 8683 | 8728 | 8774 | 8819 || 8865 | 8911 | 8956 | 9002 | 9047 
3 9093 | 9138 | 9184 | 92380 | 9275 9321 | 9866 | 9412 | 9457 | 9503 
4 9548 | 9594 | 9639 | 9685 | 9780 || 9776 | 9821 | 9867 | 9912 | 9958 
5 | 980003 | 0049 | 0094 ' 0140 | 0185 || 0231 | 0276 | 0322 | 0367 | 0412 
6 0458 | 0508 | 0549 | 0594 | 0640 0685 | 0730 | 0776 | 0821 | 0867 
té 0912 | 0957 | 1008 | 1048 | 1093 1189 | 1184 | 1229 | 1275 | 1320 
8 1366 | 1411 | 1456 | 1501 | 1547 1592 | 1637 | 1688 | 1728 | 17738 
9 1819 ; 1864 |! 1909 | 1954 | 2000 '| 2045 | 2090 | 2185 | 2181 | 2226 

960 2271 | 2316 | 2362 | 2407 | 2452 || 2497 | 2543 | 2588 | 2633 | 2678 
1 27238 | 2769 | 2814 | 2859 | 2904 || 2949 | 2994 | 3040 | 3085 | 3130 
2 3175 |. 8220 |. 38265 | 3310 | 3356 || 3401 | 3446 | 3491 | 3536 | 3581 
3 3626 | 3671 | 3716 | 3762 | 3807 || 3852 | 3897 | 3942 | 3987 | 4032 
4 4077 | 4122 | 4167 | 4212 | 4257 || 4802 | 4847 | 4392 | 4437 | 4482 
5 4527 | 4572 | 4617 | 4662 | 4707 || 4752 | 4797 |-4842 | 4887 | 4982 45 
6 4977 | 5022 | 5067 | 5112 | 5157 |} 5202 | 5247 | 5292 | 53837 | 5882 
if 5426 | 5471 | 5516 | 5561 | 5606 || 5651 | 5696 | 5741 | 5786 | 5830 
8 5875 | 5920 | 5965 | 6010 | 6055 || 6100 | 6144 |. 6189 | 6234 | 627 
9 6324 | 6869 | 6413 | 6458 | 6503 || 6548 | 6598 | 6687 | 6682 | 6727 

970 6772 | 6817 | 6861 | 6906 | 6951 || 6996 | 7040 | 7085 | 7130 | 7175 
1 7219 | 7264 | 7809 | 73853 | 73898 || 7443 | 7488 | 7582 | 7577 | 7622 
2 7666 | 7711 |.7%56 | 7800 , 7845 || 7890 ; 7984 | 7979 | 8024 | 8068 
3 8113 | 8157 | 8202 | 8247 | 8291 8336 | 83881 | 8425 | 8470 | 8514 
4 8559 | 8604 | 8648 | 8693 | 8737 || 8782 | 8826 | 8871 | 8916 | 8960 
5 9005 | 9049 | 9094 | 9138 | 9183 || 9227 | 9272 | 9816 | 9361 | 9405 
6 9450 | 9494 | 9589 | 9583 | 9628 || 9672 | 9717 | 9761 | 9806 | 9850 
fe 9895 | 9939 | 9988 |——— — 

_— —] 0028 | 0072 || 0117 | 0161 | 0206 | 0250 | 0294 
8 | 990339 | 0883 | 0428 | 0472 | 0516 || 0561 |} 0605 | 0650 | 0694 | 07 
9 0783 | 0827 | O871 | 0916 | 0960 || 1004 | 1049 | 1093 | 1137 | 1182 

980 1226 | 1270 | 13815 | 1859 | 1403 || 1448 | 1492 | 15386 | 1580 | 1625 
1 1669 | 1718 | 1758 | 1802 | 1846 1890 | 1935 | 1979 | 2028 | 2067 
Ps 2111 | 2156 | 2200 | 2244 | 2288 || 2383 | 2877 | 2421 | 2465 | 2509 
So 2554 | 2598 | 2642 | 2686 | 27380 || 277 2819 | 2863 | 2907 | 2951 
4 2995 | 38039 | 3083 | 3127 | 3172 || 8216 | 38260 | 33804 | 3348 | 3392 
5 38436 | 3480 | 3594 | 38568 | 3613 || 8657 | 3701 | 3745 | 3789 | 3833 
6} 38877 | 3921 | 3965 | 4909 | 4053 || 4097 | 4141 | 4185 | 4229 | 4278 
fe 4317 | 4361 | 4405 | 4449 | 4493 || 4537 | 4581 | 4625 | 4669 | 4713 44 
8 4757 | 4801 | 4845 | 4889 | 4933 || 4977 | 5021 | 5065 | 5108 | 5152 
9 5196 | 5240 | 5284 | 5828 | 53872 || 5416 | 5460 | 55u4 | 5547 | 5591 

PROPORTIONAL PARTS. 

Diff. 1 4) | 3) 4 5 6 7 8 9° 
46 4.6 9.2 13.8 18.4 23.0 27.6 32.2 36.8 41.4 
45 4.5 9.0 13.5 18.0 2225 27.0 3125 36.0 40.5 
44 4.4 | 8.8 13.2 17.6 22.0 26.4 30.8 35.2 39.6 
43 | 4.3 8.6 12.9 Aco 21.) 8 30.1 34.4 38.7 





TABLE XXIV.—LOGARITHMS OF NUMBERS 


€ a 












































































































































No. 990 L. 995.] [No. 999 L. 999. 
N. 0 2 3 4 5 6 % 8 9 | Diff. 
990 | 995635 | 5679 | 5723 | 5767 | 5811 | 5854 | 5898 | 5942 | 5986 | 6030 
1 6074 | 6117 | 6161 | 6205 | 6249 | 6293 | 6337 | 68380 | 6424 | 6468 th 
2 6512 | 6555 | 6599 | 6643 | 6687 | 6731 | 6774 | 6818 | 6862 | 6906 
3 6949 | 6993 | 7037 | ‘7080 | 7124 | 7168 | 7212 | 7255 | 7299 | 7343 
4 7386 | 7430 | 7474 | 7517 | 7561 | 7605 | 7648 | 7692 | 7786 | 777 
5 7823 | 7867 | 7910 | 7954 | 7998 | 8041 | 8085 | 8129 | 8172 | 8216 
6 8259 | 8303 | 8347 | 8390 | 8434 | 8477 | 8521 | 8564 | 8608 | 8652 
vi! 8695 | 8739 | 8782 | 8826 | 8869 | 8913 | 8956 | 9000 | 9043 | 9087 
8 9131 |. 9174 | 9218 | 9261 | 9305 | 9348 | 9892 | 9485 | 9479 | 9522 
9 9565 | 9609 | 9652 | 9696 | 9739 | 9783 | 9826 | 9870 | 9913 | 9957 43 
LoGAaRITHMS OF NUMBERS FROM 1 To 100. 
N. Log. N. Log. N. Log. N. Log. N. Log. 
1} 0.000000 || 21 | 1.822219 | 41 | 1.612784 |; 61 | 1.785330 |; 81 | 1.908485 
2} 0.301030 || 22 | 1.842428 || 42 | 1.623249 || 62 | 1.792392 || 82 | 1.918814 
3 | 0.477121 || 23 | 1.361728 || 43 | 1.633468 || 63 | 1.799341 || 83 | 1.919078 
4| 0.602060 || 24 | 1.880211 | 44) 1.643453 |) 64 | 1.806180 || 84 | 1.924279 
5 | 0.698970 |] 25 | 1.397940 | 45 | 1.653218 || 65 | 1.812918 |) 85 | 1.929419 
6| 0.778151 || 26 | 1.414973 | 46 | 1.662758 || 66 | 1.819544 || 86 | 1.934498 
4%) 0.845098 || 27 | 1.481364 || 47 | 1.672098 || 67 | 1.826075 7 | 1.939519 
8 | 0.903090 |} 28 | 1.447158 || 48 | 1.681241 |] 68 | 1.832509 || 88 | 1.944483 
9] 0.954243 || 29 | 1.462398 || 49 | 1.690196 || 69 | 1.838849 || 89 | 1.949390 
10 | 1.000000 || 80 | 1.477121 || 50 | 1.698970 || 70 | 1.845098 || 90 | 1.954243 
“11 | 1.041393 || 31 | 1.491362 || 51 | 1.'707570 || 71 | 1.851258 || 91 | 1.959041 
12 1.079181 || 32 | 1.505150 || 52 | 1.716003 || 72 | 1.857832 || 92 | 1.968788 
13 | 1.118943 |} 33 | 1.518514 || 53 | 1.724276 || 73 | 1.863323 || 93 | 1.968483 
14 | 1.146128 |} 34 | 1.531479 || 54 | 1.782394 || 74 | 1.869282 || 94 | 1.973128 
15; 1.176091 || 35 | 1.544068 || 55 | 1.740363 || 75 | 1.875061 || 95 | 1.977724 
16 | 1.204120 || 36 | 1.556303 || 56 | 1.748188 || 76 | 1.880814 || 96 | 1.982271 
17 | 1.230449 || 37 | 1.568202 || 57 | 1.755875 || 77 | 1.886491 || 97 | 1.986772 
18 | 1.255273 || 88 | 1.579784 || 58 | 1.763428 || 7 1.892095 |} 98 | 1.991226 
19 1.278754 || 89 | 1.591065 || 59 | 1.770852 || 79 | 1.897627 || 99 | 1.995635 
20 1.301080 || 40 | 1.602060 || 60 | 1.778151 || 80 | 1.908090 |/100 | 2.000000 
| 
Sign Sign | Value| Sign | Value! Sign | Value 
pie: in Ist ee ined | at |in3d! at |in4th| . at 
* | Quad. *| Quad.|. 180°, | Quad.} 270° | Quad.| 360°. 
i ee oO R a O ~ R ~ O 
Tan O co — O + oo — O 
EC Seto: R or) _ R _ m2) R 
Versin O R a 2R + R i O 
Cos sain: R O _ R -- O R 
Ofe We A ans fee) O — oro) + O — oa) 
Cosecsi.... R + oa) _ R — co 














R signifies equal to rad; © signifies infinite; O signifies evanescent, 





7 

















TABLE XXV.—LOGARITHMIC SINES 





463726 


.505118 
042906 
577668 
609853 
.639816 
667845 
694173 
.718997 
(42478 
164754 
. 785943 
.806146 
825451 
842934 
861662 
878695 
895085 
.910879 
. 926119 
940842 


7.955082 
968870 
982233 

7.995198 

8.007787 
.020021 
.031919 
043501 
054781 
06577 

8.076500 
.086965 
097183 
.107167 
116926 
126471 
1385810 
144953 
. 153907 
162681 


8, 171280 


+ 


-F 


8. 241855 


Cosine. 


























Tang. 


“463727 
"7 505120 
"542909 
577672 
609857 


639820 |) 


667849 
694179 
719003 
- (42484 
764761 


7.785951 
806155 
825460 
848944 
861674 
878708 
895099 
910894 
926134 
940858 


7.955100 
. 968889 
. 982253 
7.995219 
8.007809 
020044 
031945 
043527 
054809 
.065806 


8.076531 
086997 
097217 
107208 
116963 
126510 
185851 
. 144996 
1538952 
162727 


8.171328 
179768 
. 188036 
. 196156 
204126 
211953 
219641 
-227195 
284621 
8.241921 


Cotang. 











Cotang. 


Inf. p 

133 536274 
. 280244 

13. 059153 

12. 934214 
. 837304 
~758122 
.691175 
. 633183 
.582030 
.586273 


12.494880 
457091 
422828 
.390143 
. 3860180 
.332151 
. 805821 
. 280997 
.207516 
205239 

12.214049 
198845 
. 174540 
156056 
. 188326 
121292 


104901 | 


.089106 
073866 
059142 


12.044900 
.031111 
017747 

12.004781 

11.992191 
979956 
. 968055 
956473 
.945191 
934194 


11.923469 
. 913003 
902783 
892797 
883037 
873490 
.864149 
855004 
.846048 
837273 


11.828672 
820237 


1.758079 


Tang. 


q+l 

















D1” 


D1” 





| 
Cosine. 179° 


ten 
ten 
ten 
ten 
ten. 
ten 
9.999999 
. 999999 
. 999999 
. 999999 
. 999998 


9.999998 
.999997 
. 999997 
. 999996 
. 999996 
. 999995 
.999995 
. 999994 
-999993 
999993 


9.999992 
.999991 
.999990 
.999989 
.999989 
. 999988 
999987 
999986 
.999985 
999983 


9.999982 
999981 
999980 
.999979 
999977 
999976 
999975 
999973 
999972 
999971 


9.999969 


999968 
999966 
999964 
999963 
999961 
.999959 
999958 
999956 
999954 


9.999952 


9.999934 


Sine. 











NS OMmdwRaas 


89° 





COSINES, TANGENTS, AND COTANGENTS 


01 





























i Sine q—l Tang 

Z 4.685 
3600 | O | 8.241855 | 553 | 619 | 8.241921 
3660 1 249083 | 552 | 620 . 249102 
3720 | 2 256094 | 551 | 622 . 256165 
3180 | 3 263042 | 551 | 623 . 263115 
3840 | 4 269881 | 550 | 625 . 269956 
3900 | 5 276614 | 549 | 627 | .276691 
3960 | 6 283243 | 548 | 628 283323 
4020 | 7 289773 | 547 | 630 289856 
4080 | 8 296207 | 546 | 632 296292 
4140 | 9 . 302546 | 546 | 633 . 302634 
4200 | 10} .308794 | 545 | 635 | .308884 
4260 | 11 | 8.314954 | 544 | 637 | 8.315046 - 
4320 | 12 |" .821027 | 543 | 638 .821122 
4380 | 13 .3827016 | 542 | 640 .827114 
4440.) 14 . 832924 | 541 | 642 . 338025 
4500 15 . 838753 | 540 | 644 . 338896 
4560 |,16 | .344504 | 5389 | 646 | .344610 
4620 | 17 .3850181 | 539 | 648 . 850289 
4680 | 18 . 3855783 | 538 | 649 . 395895 
4740 | 19 . 861815 | 537 | 651 . 3614380 
4800 | 20 | .366777 | 536 | 653 | .366895 
4860 | 21 |. 8.372171 | 585 | 655 | 8.872292 
4920 | 22] .377499 | 534 | 657 | .377622 
4980 | 23 . 882762 | 533 | 659 . 882889 
5040 | 24! .887962 | 532 | 661] .388092 
5100 | 25 | .393101 | 531 | 663 | .398234 
5160 | 26 | .398179 | 530 | 666 | .398315 
5220 | 27 | .403199 | 529 | 668 | .403338 
5280 | 28 | .408161 | 527 | 670 | . 408304 
5340 | 29 | .418068 | 526 | 672 | .413213 
5400 | 80 | .417919 | 525 | 674 | .418068 
5460 | 381 | 8.422717 | 524 | 676 | 8.422869 
5520 | 82 | .427462 | 523 | 679 | .427618 
5580 | 338 | .482156 | 522 | 681 | .432315 
5640 | 34 . 436800 | 521 | 683 - 436962 
5700 | 35 | .441394 | 520 | 685 | .441560 
5760 | 36 .445941 | 518 | 688 . 446110 
5820 | 37 .450440 | 517 | 690 . 450613 
5880 | 38 . 454893 | 516 | 693 . 455070 
5940 | 89 | .4593801 | 515 | 695 | .459481 
6000 | 40 | .463665 | 514 | 697 | . 463849 
6060 | 41 | 8.467985 | 512 | 700 | 8.468172 
6120 | 42 472263 | 511 | 702 .472454 
6180 | 43 .476498 | 510 | 705 . 476693 
6240 | 44 . 480693 | 509 | 707 «480892 
6300 | 45 . 484848 | 507 | 710 . 485050 
6360 | 46 | .488963 | 506 | 713 | .489170 
6420 | 47.| .4938040 | 505 | 715 | .493250 
6480 | 48 | .497078 | 503 | 718 | .497293 
6540 | 49 | .501080 | 502 | 720} .501298 
6600 | 50 | .505045 | 501 | 723 | .505267 
6660 | 51 | 8.508974 | 499 | 726 | 8.509200 
6720 | 52 | .512867 | 498 | 729 | .513098 
6780 | 538 | .516726 | 497 | 731 | .516961 
6840 | 54 | .520551 | 495 | 734 | .520790 
6900 | 55 | .524843 | 494 | 7387 | .524586 
6960 | 56 | .528102 | 492 | 740 | .528349 
7020 | 57 | .531828 | 491 | 743 | .532080 
7080 | 58 | .535523 | 490 | 745 , .535779 
7140 | 59 . 539186, | 488 | 748 . 5389447 
7200 | 60 | 8.542819 | 487 | 751 | 8.548084 

ee , 4.685 

91° Cosine. g—l Cotang. 











Cotang. 


11.758079 
. 750898 


697366 
‘691116 


11. 684954 
678878 
- 672886 
666975 
661144 
644390 
649711 
644105 
638570 
- 633105 


11. 627708 
622378 
617111 
-611908 
. 606766 
601685 
596662 
591696 
986787 
581932 


11.577131 
. 972382 
567685 
563038 
908440 
- 953890 
549387 
544930 
540519 
536151 


11. 531828 
527546 
523307 
519108 
514950 
510830 
506750 
£02707 
498702 
494733 


11. 490800 
486902 
- 483039 
479210 
475414 
471651 
- 467920 
464221 
460553 
11. 456916 


Tang. 





qa+t 











Dl” 














| 
Cosine. 178° 





Sine. 88° 


9.999934 
- 999932 
. 999929 
. 999927 
- 999925 
999922 
999920 
999918 
. 999915 
. 999913 
999910 


9.999907 
999905 
. 999902 
.- 999899 
. 999897 
999894 
999891 
- 999888 
- 999885 
- 999882 


9.999879 
. 999876 
. 999873 
. 999870 
. 999867 
999864 
999861 
999858 
. 999854 
999851 


9.999848 
. 999844 
999841 
. 999838 
999884 
. 999831 
- 999827 
999824 

- 999820 
. 999816 


9.999813 
. 999809 
- 999805 
- 999801 
. 999797 
999794 
. 999790 
, 999786 
. 999782 
999778 


9.999774 
- 999769 
999765 
. 999761 
999757 
999758 
. 999748 
999744 
. 999740 

9.999735 





TOR MWRUOMAIOO 


TABLE XXV.—LOGARITHMIC SINES 


———<—<—<—$—$——$—$—— el 


| 
2° | 


0! 
1 


fat 
SO HAF Sd OT 09 0 


60! 








Sine. 


8 542819 
546422 
549995 
553539 
557054 
-560540 
563999 
“567431 
50836 
574214 
577566 

8.580892 
584193 
“587469 
590721 
593948 
597152 
600332 
“603489 

606623 
609734 


8.612823 
.615891 
618937 
-621962 
624965 
.627948 
-6380911 
638854 
686776 
. 639680 


8.642563 
.645428 
648274 
.651102 
.653911 
.656702 
.659475 
662230 
. 664968 
. 667689 


8.670893 
673080 


675751. 


678405 
681043 
683665 
686272 
.688863 
.691438 
.693998 


8.696543 
699073 
701589 
704090 
106577 
709049 
711507 
(18952 

716883 

8.718800 





Dali 


42.92 
42.67 


42.42 
42.17 





41.93 
41.68 
41.45 
41.20 
40.97 
40.75 
40.52 
40.28 


92° Cosine. D, 1". 
2 

















Cosine. 


9.999735 
.999731 
999726 
999722 
999717 
.999713 
999708 
999704 
. 999699 
999694 
. 999689 


9.999685 
999680 
. 999675 
.999670 
999665 
. 999660 
999655 
999656 
. 999645 
.999640 


9.999635 
. 999629 
999624 
.999619 
.999614 
. 999608 
.999603 
.999597 
. 999592 
. 999586 


9.999581 
999575 
999570 
. 999564 
.999558 
999553 
.999547 
.999541 
999535 
999529 


9.999524 
.999518 
.999512 
999506 
999500 
.999493 
999487 
.999481 
999475 
. 999469 


9.999463 
. 999456 
.999450 
999443 
999437 
999431 
. 999424. 
999418 
999411 


9.999404 


Sine. 


Deth 























Cotang. 


Tang. 


8.543084 
.546691 
550268 
.553817 
557386 
560828 
.564291 
567027 
571187 

. 574520 

577877 

8.581208 

584514 


Oris. 


60.12 
59.62 
59.15 
58.65 
58.20 
57.72 
57.27 
56.83 
56.38 
55.95 
55.52 


55.10 
54.68 





587795 
‘591051 
594283 
597492 
600677 
603839 
606978 
610094 


8.613189 
.616262 
.619813 
622343 
625852 
628340 
631308 
634256 
.637184 
.640093 


8.642982 
645853 
648704 
.651537 
654352 
657149 
659928 
.662689 
665433 
-668160 


8.670870 
.673563 
676239 
. 678900 
681544 
684172 
686784 
.689881 
.691963 
694529 


8.697081 
.699617 
702139 
.104646 
707140 
. 709618 
- 712083 
. 714534 
716972 

8.719396 


54.27 
53.87 
53.48 
53.08 
52.70 
52.32 
51.93 
51.58 


51.22 
50.85 
50.50 
50.15 
49.80 
49.47 
49.13 
48.80 
48.48 
48.15 


47.85 
47.52 
47 22 
46.92 
46.62 
46.32 
46.02 
45.73 
45.45 
45.17 


44.88 
44.60 
44.35 
44.07 
43.80 
43.53 
43.28 
43.03 
42.77 
42.53 
42.27 
42.08 
41.78 
41.57 

















_ Cotang. 


11.456916 
-453309 
449732 
.446183 
442664 
489172 
.485709 
432278 
428863 
425480 
422123 


-418792 
.415486 
.412205 
.408949° 
.405717 
.402508° 
.899323 
.396 161 
.3893022 
.889906 


11.386811 
.383738 
. 380687 
3877657 
.374648 
371660 
. 3868692 
865744 
. 362816 
. 809907 


1 


— 


11.357018 | | 


11 .329130 


805471 
11.302919 
3800383 
297861 
295854 
292860 
290382 
.287917 
. 285466 
283028 
11.280604 


Tang. 


17° 


60’ 








© 
S apee aera: 

































































| 
, otang. 176° 
i Dyt? Tang. | D.1". | C 
i DP iy Cosine. wats i 
3°; Sine. np he: é 
; jae 8.719396 40.17 11 sae 
-sss00 | , pi 
i 9.999404 10 eee 40.47 isin | 50 
1 : (21204 39.85 999391 oe 724204 20.73 : oie 
2 . 23595 39.62 ; 999384 "10 T6088 30.52 : 2041 
8 725972 39.42 |/- “999878 -10 Paso 39.80 ar 
4} 728387 | 39°19 999371 a ‘BIH | 39.10 ae 
S| eae | Bs oompr | oR 735996 | 38 68 261683 | 52 
6 . 1338027 38.7 999357 45 “735906 38.68 ies 
q : 735854 38.55 999350 "12 sa? 28.48 rt | Bi 
Be eee say || Sonnten a || AO | eer |: ak 
ba | pi 11 254793 
10 | .742259 | 35 os 9.999329 12 pager oes ai. 81 250 : 
é we "7. : 
12 . 746802 37 55 999315 oe 749740 srs : 21 | 4b 
13 . 749055 Sl .o8 999308 oe s7B1OG 37.0 : zat | a 
| oe | S| a 40 || TeteB NI Ba geil 241332 | 43 
Ree. be | a ig || 60872 | 36°s5 236935 | 41 
ib) roel | $8 | ae 1p || 763065 | 3¢'38 "934754 | 40 
| ie | Se ‘goats | “22 165246 | 36778 | - 3 | 39 
Bil Scheie Bo: 23 || .990265 | -22 creay| no ; 
: of 5. 
| a ae 9.999257 | “49 FOAL | 90.0 Sot 
22 | .768828 35.70 || “gq9049 iy TO | 5 Si 
zi) eaiot | P| go aD a perrepo| Beene | oe 221886 | 34 
a5 | “ines | 3-2 ey 12 778114 | 35°93 (219778 | 33 
25 | .VTS228 | gee | ‘gospan | fant 2 area 
zg fet) oe ‘aooans | “22 782320 | 34°80 "915592 | 31 
av | .779484 | 34 °g3 ‘g99205 | 22 | 9p aoe 
a5 | pi 34.50 “go9189| 13 || “486486 ig, | See 
ef : ! 
0 ata 25 iets a 8.788554 34.32 Sie 2 
: eff 3 
Bier | Hs ‘gone | 23 792062 33:96 pee 26 
eames) Bap | sane - -800763 | 33°37 "197235 | 22 
“RE os Oar ae ee 33 |) AEB) Bae 138 | § 
39 A fi 39. fi 13 3 
40 ; jaa 32.78 Gates oe s e087 2.7 7 ie : 
Bee eat, aares learn i) #8 |) “ston a | jee 
eer arco Fa pall! aoe |) BUEN | 395 ea 
46 ; 817522 31.90 999053 45 820884 31.90 tes | 
48 2 en a ‘ a , eee “45 Aces 31.63 f int | 10 
| a1 eee si 7992 7 | 1.172008 | 9 
Bi como ore 9.999019 15 || 8-827992| 91 5 17200 
* 829874 | 54°54 126 | 8 
51 | 8.827011 Sis 22 -999010 13 a 51.23 isa 
52 | 828884 | 310g “999002 7 SITS | os ast | 6 
—=53 . 880749 30.97 998993 Ge 838618 30.9% is 
54 .832607 80.82 |) _ 998984 43 foot 30.88 ite) 4 
56 . 836297 30.55 998967 ae 839163 30.58 fa 
57 | 838130 | 39°43 998958 "13 ‘siames | 20-45 ts | 
58 .839956 30.30 ae 13 9a coals 
: 59 3 841774 30.18 9.998941 | sca Ne — 
= 1. aie Da Cotang. | D. 1". | 
93° i ce a Sine. ees 
93° Cosine. 3 











4°} Sine. 
0’| 8.848585 
1 | .845387 
2} .847183 
3} .848971 
4} .850751 
5 852525 
6 | .854291 
7 | .856049 
8 | .857801 
9 | .859546 
10 | .861283 
11 | 8.863014 
12 | .864738 
13 | .866455 
14 | .868165 
15 | .869868 
16 | .871565 
17 | = .8%8255 
18 | .874938 
19 | .876615 
20 | .878285 
21 | 8.879949 
22 | .881607 
23 | .888258 
24 | 884903 
25 | .886542 
26 | . 888174 
27 | .889801 
28 | .891421 
29 | .893035 
80 | .894643 
31 | 8.896246 
82 | .897842 
83 | .899432 
34 | .901017 
35 | .902596 
36 | .904169 
37 905736 
88 | .907297 
39 908853 
40 | .910404 
41 | 8.911949 
42 | .913488 
43 | .915022 
44 916550 
45 918073 
46 919591 
47 | .921103 
48 | .922610 
49 | .924112 
5U0 | 925609 
51 | 8.927100 
52 | .928587 
53 | .930068 
54 | .981544 
55 | .988015 
56 | .934481 
57 | .985942 
58 | .9387898 
59 | _.938850 
60’| 8.940296 








94°] Cosine. 


TABLE XXV.—LOGARITHMIC SINES 


D. 











D. 


ake 











Cosine. 





9.998941 
998932 
998923 
.998914 
998905 
- 998896 
998887 
998878 
998869 
998860 
. 998851 


9.998841. 
998832 
“998823 
-998813 
“998804 
“998795 
998785 
“99877 
"998766 
(998757 


9.998747 
. 998738 
998728 
.998718 
998708 
.998699 
. 998689 
998679 
. 998669 
. 998659 


9.998649 
. 998639 
. 998629 
. 998619 
. 998609 
998599 
. 998589 
998578 
. 998568 
998558 


9.998548 
998537 
. 998527 
. 998516 
998506 
998495 
998485 
998474 
998464 
998453 


9.998442 
998431 
998421 
. 998410 
9983899 
998388 
998377 
. 998366 
998355 

9.998344 


Sine. 








DEAN 








Tang. Dras 
8.844644 
"846455 peat 
848260 | 50’ gs 
850057. | 0 'g0 
851846 | 59'n9 
853628 99 58 
855403 | $a 'ge 
BTITL | Baas 
'858982 | Soop 
860686 | 5a"45 
"862433 
8.864173 er 
86417 
965006 | 38-88 
1867632 | 58" gs 
‘egous1 | Sere 
(871064 | 59°43 
872770 | 58'90 
1874469 | 58°95 
severe? |) 304s 
877849 | 38°00 


8.881202 2 
agag69 | 20-48 
984530 | 27-68 
"886185 
-887833 | 97" 38 
$0112 
"ggnz4o | ee. 
“g94366 | 27-08 
See: | 90.0% 
26.87 

8.897596 : 
“goge03 | 26.78 
900803 | 26-67 
"902398 

R398 | 96.48 
903987 | 36-48 
"905570 
"907147 
‘gosvi9 | 26-20 

26.10 
"910285 
Pee |) 9808 
25 92 


8.913401 
"914951 | 22-88 
25.73 
916495 | 52°k3 
18084 
"919568 
‘g21096 | 22-47 
21 25 38 
"924136 
“925649 | 22: 
5 25.12 
8.928658 | oy os 








931647 | 24-8% 
24.7 
"933134 ‘ 
94." 
934616 
24 62 
"936093 
6 2453 
987565 
24.45 
"939032 
24.37 
94044 | 24:37 
8.941952 


Cotang. | D.1". 











| 
Cotang. 175° 


11. 155356 
. 153545 
151740 
149943 
-148154 
- 146372 
144597 
142829 
141068 
1389314 
137567 


11.135827 
134094 
182368 


130649 | 


. 128936 
127230 
125531 
- 128838 
122151 
120471 


11.118798 
117131 








115470 © 


118815 


112167 


. 110524 


108888 | 


107258 
105634 
104016 


11. 102404 
100797 
.099197 


097602 | 
096013 | 


.094480 
092853 
091281 
089715 
088154 


11086599 


085049 


088505 
.081966 
080432 
. 078904 


077381 | 


075864 
074851 
072844 


11.071842 
069845 
068353 
. 066866 
065384 
068907 
062485 
060968 

059506 


11.058048 


Tang. 








COSINES, TANGENTS, AND COTANGENTS 





5°| Sine. 


0’| 8.940296 


1| 941738 
2] 1943174 
3 | .944606 
4| 946034 
5 | .947456 
6) 948874 
7 | .950287 
8 | .951696 
9 | 953100 

10 | .954499 

11 | 8.955894 

12 | .957284 

13 | .958670 

14 | .960052 

15 | .961429 
16 | .962801 
17 | .964170 

18 | .965534 
19 | .966893 

20 968249 

21 | 8.969600 
22) 970947 
23] 972289 

24 | 973623 

25 | .974962 
26. | .976293 
a7 | 977619 
23 | 978041 
29 | 980259 
30. | .981573 

31 | 8.982883 
32} .984189 

33. | 985491 
34 | .986789 

35 | .988083 

36 | .989374 

37 | .990660 

38 | .991943 

39 | 993222 

40 | .994497 

41 | 8.995768 

42 | 997036 

43 | 998299 

44 | 8.999560 

45 | 9.000816 

46 | .002069 

47 | 003318 

48 | .004563 

49 | .005805 

50, .007044 

51 | 9.008278 

52 | 009510 

53. | .010737 

54 | .011962 

55 | .013182 

56 | .014400 

57 | .015613 

58 | .016824 

018031 





HO. | 26 
60’ | 9.019235 


95°| Cosine. 














Dats. Cosine. 
9.998344 
24.03 || “ "998333 
33.93 || 998822 
23,80 998311 
23.80 || 998300 
_ 3.7) || “998280 
23.03 ||  '08e77 
33.05 || 998266 
23.48 || 998255 
33.20 || 998243 
3.% "998232 
23.25 || oon 
ie 9. 
33-17 || 1998209 
23.10 || 1998197 
83.03 || 1998186 
Bee || 998174 
33-87 || 998163 
22-82 || “998151 
22-73 || 998139 
22.65 || 1998128 
83-00 || “998116 
ie 9.998104 
i 
ee ay || 998092 
33-37 ||: 998080 
83°88 || 998068 
33-83 || "998056 
23.18)) 998044 
32-10 || '998032 
33.08 || 998020 
21.97 || 998008 
21.90 || “997996 
21.83 
21,77 || 9-297984 
21 7 997972 
21.42 || 997959 
21.68 || -agvos7 
21.57 |) .997935 
21.52 || ‘agvoee 
31.43 || 997910 
31.38 || “og7B07 
21.32 || “997885 
-25 || “gove7e 
ime 9.997860 
- 997 
ae 997847 
ee? || 907885 
21,02 || “997822 
ee "997809 
20.88 || g9v797 
30.82 || ‘07784 
20.03 || 997771 
ay oe || -997758 
ae | eee 
30.53 |] “997719 
apap || -997706 
30a || 997693 
30-33 J| 1997680 
2 ee | 997667 
3.22 || 997654 
30.18 || 907641 
By oF || 997628 
9.997614 
D. 1" Sine. 














Dis 








95 


— re! 


| 
Tang. | D.1". | Cotang. 174° 





8.941952 | 94 99 
943404 | 24-20 
944852 


‘uoa0s | 28 
. 947734 ; 
: 23.90 
: 6 cs 
23.82 
. 95059 23 73 
. 952021 BS" 
23.67 
.95344 : 
23.58 
. 954856 93 5D 
. 956267 ; 


DOROT | 88.45 
8.957674 23.35 
900075 | Be « 

ar Ares 23.30 
.960473 2 6 
» | 23.22 
. 961866 23 15 
. 968255 eat 


"964639 ae 
966019 | 33-00 
967304 | 3o-02 
"968766 





Ear ales mat 

. uae 99 7% 
S7OT1800 "| 60.66 
.972855 Dy) 57 
974209 | G52" 


‘975560 | 22-52 

ff 22.43 
a 94 6906 oO 37 
978248 “en 
“Oboe | 22-25 


me 22.19% 
Se 22.10 
2203 

8.984899 


986217 21.97 
“987532 | SI: 
ogesd2 | 31-88 
990149 1 

991451 | 3t° 

992750 | Sine 
994045 | 54°P5 
995837 | S1-% 
996624 | 51° 


8.997908 

8.999188 Aron 

9.000465 | 31-28 
001738 | 31° 








"003007 
‘oodg72 | 22-08 





005534 | 31°03 
006792 2092 
008047 | S>: 

Gansas 20:88 
Se 20.80 


9.010546 rt 
‘ies | 38 
(014268 | 20-62 


‘015502 | 20-57 
2 | 90150 
016732 
20.45 
017959 
019183 | 20-40 
20.33 


020403 











9.021620 | 20-28 





Cotang. Di: 





11.058048 
056596 
} .055148 
053705 
052266 
-050832 
-049403 
047979 
046559 
* .045144 
. 043733 
11042326 
040925 
039527 
038134 
036745 
035361 
033981 
032606 
031234 
029867 
11028504 
-027145 
025791 
.024440 
023094 
.021752 
020414 
.019079 
017749 
016423 


11.015101 
013783 
012468 
011158 
009851 
908549 
007250 
005955 
.004663 
003376 

11.002092 

11.000812 

10.999535 
998262 
. 996993 
995728 
.994466 
993208 
.991953 
. 990702 


10.989454 
988210 
. 986969 
985732 
. 984498 
- 983268 
982041 
980817 
- 979597 
10.978380 











Tang. 





dO CO OL 3 0 


or 
~ 


84° 





96 


6° 


0’ 


SMWwWWD WHY He ee a ad 
ri rd SSRSRRLNRNN SWWAIMAOE WMH DMNOIMHMUPRWWH 


59 


60’| 9.085894 


96°} Cosine. 














Sine, 


9.019235 
020435 
0216382 
022825 
.024016 
025203 
.026386 
027567 
028744 
-029918 
.031089 


9.032257 
033421 
-034582 
035741 
. 036896 
.038048 
.039197 
.040342 
041485 
042625 

9 043762 
044895 
.046026 
047154 
648279 
.049400 
050519 
051635 
052749 
053859 


9.054966 
056071 
057172 
058271 
059367 
.060460 
.061551 
062639 
068724 
064806 


9.065885 
066962 
. 068036 
.069107 
070176 
071242 
072806 
.073866 
074424 
075480 

9.076533 
077583 
078631 
079676 
080719 
081759 
082797 
083832 
084864 











TABLE XXV.—LOGARITHMIC SINES 


D. 1". 





Cosine. 


9.997614 
.997601 
997588 
997574 
997561 
997547 
997534 
997520 
997507 
997493 
997480 


9.997466 
997452 
997489 
997425 
997411 
997397 
-997383 
997369 
997355 
997341 


9.997827 
.997313 
997299 
997285 
997271 
997257 
997242 
997228 
997214 
.997199 


9.997185 
997170 
997156 
997141 
997127 
997112 
.997098 
997083 
997068 
997053 

9.997039 
997024 
- 997009 
996994 
. 996979 
996964 
. 996949 
. 996934 
996919 
.996904 


9.996889 
996874 
. 996858 
996843 
996828 
996812 
996797 
996782 
| .996766 
| 9.996751 














Sine. 


Da" 











Tang. 


9.021620 


022834 
024044 
025251 
.026455 
027655 
028852 
030046 
031237 
032425 
033609 
9.034791 
-035969 
037144 
038316 
.039485 
040651 
041813 
042973 
.0441380 
045284 


9.046434 
047582 
048727 
049869 
-051008 
-052144 
053277 
054407 
055535 
056659 


9.057781 
058900 
.060016 
.061130 
062240 
063348 
064453 
965556 
066655 
067752 


9.068846 
069938 
071027 
072113 
073197 
074278 
075356 
.076432 
077505 
078576 


9.079644 
080710 
081773 
082833 
083891 
084947 
.086000 
087050 
088098 

9.089144 


Cotang. 











Da 


20.23 
20.17 
20.12 
20.07 
20.00 
19.95 
19.90 
19.85 
19.80 
19.73 
19.70 
19.63 
19.58 
19.58 
19.48 
19.48 
19.37 
19.38 
19.28 
19.23 
DOR, 


19.13 
19.08 
19.03 
18.98 
18.93 
18.88 
18 83 
18.80 





10.978380 
977166 
975956 
974749 
973545 
972345 
-971148 
969954 
968763 
. 967575 
-966891 


10. 965209 
964031 
. 962856 
. 961684 
. 960515 
-959349 
9581 37 
957027 
955870 
954716 


10. 953566 
952418 
-951273 
- 950131 
- 948992 





947556 
946723 
945593 
- 944465 
943841 


10 .942219 


10.931154 
930062 
928973 
927887 
926803 
925722 
924644 
923568 
922495 
921424 


10920356 





-919290 
918227 
917167 
.916109 
.915053 
- 914000 
-912950 
911902 


10.910856 


Tang. 








i 
Cotang. 173° 


QHWW RAR IAO 


83° 





COSINES, TANGENTS, AND COTANGENTS 








7? Sine. 
0’) 9.085894 
1 .086922 
2 087947 
S .088970 
4 .089990 
5 .091008 
6 .092024 
ae .093037 
8 .094047 
9 .095056 

10 .096062 

11 | 9.097065 

12 .098066 

13 .099065 

14 .100063 

15 .101056 

16 . 102048 

17 . 103037 

18 . 104025 

19 . 105010 

20 .105992 

21 | 9.106973 

22.) 107951 

23 . 108927 

24 .109901 - 

25 .110873 

26 .111842 

27 .112809 

28 11377 

29 .114737 

30 .115698 

31 | 9.116656 

32 .117613 

33 . 118567 

34 .119519 

35 . 120469 

36 .121417 

37 . 122362 

38 . 123306 

389 |  .124248 

40 . 125187 

41 | 9.126125 

42 . 127069 

43 . 127993 

44 . 128925 

45 .129854 

46 .130781 

47 .131706 

48 .132630 

49 6133551 

50 134470 

51 | 9.135387 

52 | .136303 

53 .187216 

54 . 188128 

55 . 189037 

56 . 189944 

57 .140850 

58 141754 

59 . 142655 

60’| 9.143555 








97°| Cosine. | D.'1". 




















Cosine. 


9.996751 
.996735 
996720 
996704 
.996688 
. 996673 
. 996657 
.996641 
. 996625 
. 996610 
996594 


9.996578 
. 996562 
996546 
. 996530 
996514 
.996498 
. 996482 
. 996465 
-996449 
. 996433 


9.996417 
. 996400 
. 996384 
. 996368 
996351 
996335 
-996318 
. 996302 
996285 
. 996269 


9.996252 
. 996235 
- 996219 
. 996202 
996185 
. 996168 
.996151 
-9961384 
.996117 
. 996100 


9.996083 
996066 
. 996049 
. 996082 
- 996015 
. 995998 
. 995980 
. 995963 
. 995946 
995928 


9.995911 
995894 
995876 
.995859 
995841 
995823 
995806 
995788 
99577 

9.995753 








Sine. 




















Tang. 


9.089144 
090187 
091228 
.092266 
093802 
.094336 
095867 
.096395 


* 097422 


098446 
099468 


9.100487 
. 101504 
102519 
. 1038582 
. 104542 
. 105550 
. 106556 
107559 
. 108560 
109559 


9.110556 
111551 
112543 
118533 
114521 
115507 
116491 
117472 
118452 
119429 


9.120404 
121877 
122848 
1238317 
124284 
125249 
126211 
127172 
128130 
129087 


9.180041 
180994 
181944 
182893 
. 138839 
134784 
185726 
186667 
. 137605 
188542 


9.139476 
140409 
141840 
142269 
143196 
144121 
145044 
145966 
. 146885 

9.147803 





Cotang. | D. 1". 











J 
Cotang. 172° 


10.910856 
909813 
908772 
907734 
- 906698 
905664 
904633 
- 903605 
902578 

» .901554 
- 900532 


10.899513 
898496 
.897481 
.896468 
.895458 
894450 
.898444 
892441 
.891440 
.890441 


10.889444 
"888449 
887457 
"886467 
“885479 
"884493 
"883509 
882528 
881548 
880571 


10.879596 
878623 


870913 


10. 869959 
869006 
868056 
.867107 
.866161 
865216 
864274 | 
863333 
862395 
,861458 


10. 860524 
859591 
. 858660 
857731 
. 856804 
855879 
854956 
8540384 
853115 
10 .852197 


Tang. 





| Ord COR OV IDO 











98 TABLE XXV.—LOGARITHMIC SINES 
s°| sine. | v.1 || cosine. | p.1% || Tang. | p. 1". | Cotang. 171° 
Spubitnaes als ira | 
0/| 9.143555 9.995753 9.147803 “60 
1) 144453 ate 990735 “oe 148718 HS TO eeiase | £9. 
3 | cligess | 14-90 || “goseag | -80 Toei | 15-20 |> ogagape Hee 
3) 143 | 14-88 || -Gosuer | 82 |] “dbiasa | 15-17 | “pannan’ | op 
f | Teepe | 14:83 |) “Bopee4 | 28 |] “aea6a | 15-15 |  -Barear | be 
o | ctaoie | 24-82 |! conseag | -20 |! tsae69 | 15-10] “Stored | ba 
Beane | SAS’ common | 180.. || aeaeenea | BAS |! Seana tee 
gy epsee | 14°73] Bosse | 82 || ctgsorz | 15-05 | “Bagpps | b2 
9 | aeise9 | 24-72 || “epss cg |] Clee Age iee | eee 
10 i 59 4 14.7 : 5091 30 155978 14.98 844022 | 51 
11 “sien M00 |e NBO od ey 
1/ 9. 9.995555 9.157775 
1 | "154208 ee 999537 00 || segtoeers dara Oet309 | 48 
: : . i 159565 : 840435 
14| 1155957 | 14-57 || ‘oo5501 | —-32 160457 | 14-87) * fh 
15 | .156830 | 14:55 || ‘g9s4ga .B2 361317 | 14-88 .839543 | 46 
t6| cisrroy 144-50 || “g9s46a | -20 |! “joa036 | 14-82]  “Rarred | ag 
17 158569 14.48 . ( .30 . 14.78 887764 44 
test geeaee | 4 laces | bee? |l Eeeaanne tains | 886877 | 48 
19 . 160801 14.43 "99540 30 Pasig 14.73 . 835992 | 42 
Bo teties | 14:88 || <Spsngo | 382. | cgesera | 14-20 | ce -geane lan 
a1 | 9.169098 | 1, || 9.998872 | uke | Papen ee 
162025 99537 9.166654 33: 
22162885 | 15 || 90553 |g || 167532 | 1G 10-890108 | 38 
a (ARES | ew | ORR |) eo | AEE | ds | re de 
25 | .165454 23 || “995097 | 82 vei || OEDEO i ae 0716 | 36 
26.| 166307 | 14-22 || ‘99597 82 | eae 14.53 | -829848 | 85 
or | “i67is9 | 14:20 | “gosa60 | -20 || “trasao | 24-50 | “Rotor | 38 
98} (168008 | 14-15 || ‘g95047 | — -32 syeray | at -828101 | 33 
99 | “16856 | 14-13 || ‘gg5s02 | -32 || “i736 14.45 | 827233 | 82 
| SRabBros | A410 Ht egress | eB. || egyanng | SPSH2 | eee 
gL waeeesay 14.08 || - "82 174499 | 44°59 .825501 | 30 
1) 9.1% 9.995184 9.175362 . 
es | scanen if oe ae ay || SeAYBBOd | Tee Dedearre! ee 
34 178070 | .d7y> “995127 os “irrote 14. 30 | “Boone | 36 
.173908 995108 | °° “a7e799 | 14-28) — “go401 
36 174744 13.93 995089 By : 1796 14 97 .821201 | 25 
eerie 3. - JIU : 55 820845 
37 | (175578 | 18-90 |) ‘og5070 | (182 Tansee | See am 24 
ee ¢ . 08 
38 | 16411 | 18-88 82 14.99 | -819492 | 28 
39 | 177242 | 18-85 acon 82 -181360 | 447g}  -818640 | 22 
ol taeepos |! ABER || oteeenee | 82 Hl icamenty Cees cegngy ana 
41 | 9.178900 18.80 |f ee | 88 188059} 44/13 | _-816941 | 20 
1789 wy || 9.994993 9.183907 
4.119726 ie 94074. se | Oorretvcel ales 70 -g15248 | 18 
180551 | 33" || 1185597 | 44. 814403 
44| 1181374 | 18-72 || ‘go4995 | — «38 ease. | 21408) ae 17 
de iieteeios: | 28.20 ll eteeapia: O82 ||| agimrene i eet Iegaeeges are 
99491 1187280 | 34. 812720 
46 | .193016 | 13-87 || “g94896 | «38 14.00 | 812720 | 15 
E : .188120 ; 811880 
47 | |198834 | 13-63 || “go4ar7 | «82 13.97 | - 14 
. . 188958 : 811042 
48 | .194651 | 18-82 || “og4e57 | — -38 958 | tig 93 |: 13 
13.58 || ° -189794 .810206 
SF lati veeece |) bey || oneeaes 33. || 190629 | 73-82]  's09371 it 
st | 9.187092 | 1. <> ey (eS. nee | MBN ee 
187 9.994798 9.192294 ro 
be | aprons | 848 | °-soareo | 2 |) aaoiat | 88 | sore | 
2 . (1d q Ur p . 
tees | ee ae] oe | ee | dee) eee 
56 | .191130 | 18-42 || ‘o94700 | -33 || 196 Hic mie cieg 6) 
s ; - 196430 803570 | 4 
57 | '191933 | 13-88); “oo4eg0|  -22 sorocs | 18:t Bi 
Pilletataees || 18:85 |] Uiemeeay | 88. | acauaeen ad) OMUR IN mania ct tae 
Bh Gidnases | 18:88 || mtepmant "83 || 2 genena) 80) coentioe: ime 
: 13.30 || g° "33 -198894 801106 | 1 
60" | 9.194382 | 9.994620 9.199713 | 13-6 | i9 gooe87 | 0 
98° Cosine. | D. 1’ Sine. D.1" Cotang. | D. 1’. Tang. 81° 

































































COSINES, TANGENTS, AND COTANGENTS 


9° Sine. Dai" 


~ 


195129 
195925 
.196719 
199001 a 


~ 200666 


OMIMDUP WMH OS 


Cosine. 


9.194332 9.994620 
oe 994600 


197511 a qe {|  .994540 
25 13.18 ; 
198302 1315 .994519 
.199879 13.12 994479 
.201451 13 05 . 9944388 


10 | .202284 13.05 994418 


11 | 9.208017 | 45 9 || 9.994898 
994377 

13.00 || “og4357 

12°95 || - 


14 | .205354 12.95 994836 





"994316 
i3'ee ||. 994295 


. 994274 


18 | 208452} 55°63 || 994254 


19 | .209222 


1293 || -994288 


20 | .209992 | 39°99 || - 994212 
21 | 9.210760 | 49 rt 9.994191 


22 | 211526 | y5'ss || .994171 
23 | .212201 | 35°» 994150 
24 | .213055 | 35'n, 994129 


aed 
25 | 213818 | 32-42 || ‘904108 


26 | .214579 | 35'¢s-||  -994087 
27 | 1215838 | 45 gs || -994066 
28} .216097 | 45'¢ || -994045 
29 | .216854 | 35'ng || .994024 


30 | 217609 | 12.57 . 994008 


81 | 9.218363 12.55 9.993982 
82 | .219116 | 45753 . 993960 


33 | .219868 12.50 


. 9939389 


34 | .220618 12.48 993918 
35 | .221867 | 40" 47 . 993897 
86, 222115 | 49°43 993875 


37 | .222861 





. 998854 


38 | .228606 12.38 . 993832 
39 | .224849 | 45'9¢8 993811 
40 | .225092 | 49/35 993789 


41 | 9.225833 | 40 99 9.993768 
42 | .226573 12.30 .993746 


43 | .227311 





993725 


44 »228048 | 49 97 993703 
45 | 228784 3993 993681 
46 | .229518 12 93 . 993660 





7 .230252 | 39/909 . 993638 


48 | .230984 | 49°78 993616 
49 | .231715 | 39°45 . 993594 
50 | .282444 | 49°43 993572 


51 | 9.283172 | 40 49 9.998550 
52 | .238899 | 35°49 993528 
53.|  .234625 | 49g 993506 
54 | .235849 | 4997 993484 
55 | .286073 | 49" 93- - 993462 





or 
oO 


286795 12/00 993440 


57 | .2387515 | 49°99 998418 
58 | .238235 | 4497 993396 
238953 | . 4495 993374 





59 : 
60’| 9.239670 
‘99° Cosine. | Der, 





Sine. 














DA" 





DSi 














Tang. | D. 1". 
9.199713 | 45 69 
. 200529 13.60 
.201345 13.57 
.202159 22 


202159 | 13°53 
202071 | 343 59 
203782 | 13-52 
204502 | 13.50 
"205400 | 13-47 
13.45 
306207 
6 13.43 
.207013 13.40 
207817 au 
9.208619 
"999420 | 18-85 


"210299 | 18.38 
13.30 

211018 | 13-52 

"911815 


.212611 ; 
"213405 | 18.28 
. 214198 
. 214989 i 

rs 13.18 


9.216568 | 45 43 
217356 | 13-18 
asi 10 
918926 
‘ai9710 | 13-0¢ 


920492 oe 
gerere | 12-00 
‘920052 | 13- 


12.97 
"929830 
"903607 | 12-95 








9.224382 bt 
"995156 wan 
225029 | 45 on 
226700 | 45"9° 
BRTAT1.| 45°35 
228239 | 45°49 
(229773 19.67 
230539 12°72 
"931302 bs 
en 12.72 

9.2 
“o3ag08 | 12.68 

12.67 
289586 | 55 es 
"934345 ma 
"935103 a 
235859 | 45" R3 
(236614 | 35 'ep 
.237368 | 39° 53 
288120 | 45'p5 
onl 12.50 

9.23962 
940371 abide 
241118 | 45°45 
241865 | 45°49 
242610 | 45°49 
249854 | J5'9e 


"944097 : 
"o44g39 | 12.37 
O45 


9.246319 | 1°-38 
Cotang. | D. 1’. 





99 


| 
Cotang. 170° 





10, 800287 
799471 
. 798655 
797841 
797029 
. 796218 
795408 
794600 
193793 
. 92987 
792183 


10.791881 
. 790580 
. 789780 
. 788982 
. 788185 
(87389 
. 786595 
«785802 
. 785011 
(84220 

10.783432 
782644 
. 781858 
(81074 
. (80290 
779508 
178728 
777948 
TAG 
«776393 

10.775618 
774844 
C74071 
. 773800 
72529 
A761 
770993 
T0227 
. 769461 
. 768698 


10.767935 
» 767174 
766414 
765655 
7164897 
764141 





763386 | 


762682 
. 761880 
- 761128 


10.760378 
759629 
. 758882 
- (58135 
. (57390 
. 756646 
755908 
. 755161 
754421 





10.753681 


Tang. 


wn ne a 








') ; 





100 


TABLE XXV.—LOGARITHMIC SINES 





10° 


10 








59 
60’ 


Sine. ° 


9.239670 


. 240386 
-241101 
241814 
242526 
248237 
243947 
244656 
245363 
. 246069 
246775 


9.247478 
248181 
248883 
249583 
250282 
200980 
251677 
202373 
208067 
258761 


9.254453 
255144 
200834 
206523 
207211 
257898 
258583 
259268 
209951 
260633 


9.261314 
.261994 
262673 
268351 
264027 
-264708 
265377 
266051 
266723 
267395 


9.268065 
268734 
- 269402 
-270069 
270735 
-271400 
272064 
2202726 
273388 
274049 


9.274708 
275867 
276025 
276681 
277337 
207991 
248645 
219297 
279948 

9.280599 





Dai 





100°Cosine. | D. 1’. 





Cosine. 





9.993351 
993329 
993307 
993284 
993262 
993240 
993217 
.993195 
.993172 
. 993149 
993127 


9.993104 
.993081 
- 993059 
993036 
.993018 
. 992990 
992967 
992944 
992921 
. 992898 


9.992875 
992852 
. 992829 
. 992806 
992783 
992759 
. 992736 
992713 
- 992690 
. 992666 


9.992643 
992619 
992596 
992572 
992549 
- 992525 
992501 
992478 
992454 
992430 


9.992406 
. 9923882 
992359 
- 992835 
.992311 
992287 
9922638 
992239 
992214 
- 992190 


9.992166 
992142 
992118 
992093 
992069 
992044 
992020 
-991996 
nQOLOCL 

9.991947 











| Sine. 























D. 1’. Tang. | D.1’. 
9.246319 
ef 947057 eae 
vk i] qeaammed | 32-85 
oe |t saeaeago | 12.8% 
Sf, || qepdeaes |) 452° 
ee Hl eeeoaren |) a2) 
ah || wepbiabt 1) ee 
8 lL ocpaatot (aga 
oe IT SePueeD Ie tue 
"253648 Hi 
"38 one 12.10 
254374 
| Seo | 29 
at veppens | 12-07 
BS Ih oabeae fee 
Be |p 2oree9 | 48 
"38 257990 | 49°00 
ae >] gueoerid | Geccs 
ae || epnedee | tee 
"3g || 260146 | 4495 
"38 ona 11.92 
9.26157 
He 962298 | = ot 
a8 || |. 36n005 | LEB 
38 268717 | 33" 85 
BES Ih Ridges || ATE 
ae | 26138 | ie 
"gg || 205847 | 44 °g9 
"38 || 206555 | 44 "an 
a Il oROTRGL |p Eee 
. 267967 
BB | ee, | Ue 
3 4 lod 
a "969375 1 ee 
ae lbegecoory | Mee 
ae arog | 11.0 
Seer tee eee 
ty, NW eeeeet7s | yates 
973573 
40 a 11.60 
al eeeraped | ree 
9.275658 slat 
ee 276351 a1 
40. Tc oerersa sh ae 
ao organs | 11-58 
oi erogo1 | 41-47 
ae: || saeelted | Aas 
40 |) peti 11.40 
28255 
“40 || 1288225 ae 
te || On eaBOeT | Rae 
"40 286624 | 47/98 
ee 287301 | 11.28 
an 2B007F |) hs a 
9. 288652 


D. 1". Il Cotang. | D. 1". 


i 
Cotang. 169° 


10.753681 | 60’ 
. 752943 | 59 
. 752206 | 58 
751470 | 57 
750736 | 56 
750002 | 55 
.749270 | 54 
748539 | 53 
747809 | 52 
747080 | 51 
746352 | 50 


10.745626 | 4% 
744900 | 48 
744176 | 47 
748453 | 46 
742731 | 45 
742010 | 44 








741290 | 43 
740571 | 42 
7389854 “| 41 
. 739137 | 40 


10.738422 | 39 
737708 | 38 
736995 | 37 
. 7386288 | 36 
785572 | 35 
734862 | 34 
784153 | 33 
738445 | 382 








732739 | 31 
.782033 | 30 


10.7313829 | 29 
(30625 | 28 
429923 | 27 
(29221 | 26 
728521 | 25 
- (27822 | 24 
727124 | 23 
(26427 | 22 
725731 | 21 
- 725036 | 20 


10.724342 | 19 
. 728649 | 18 
. (22957 | 17 
. (22266 | 16 
(21576 | 15 
720887 | 14 
720199 | 13 
719512 | 12 
718826 | 11 
- 718142 | 10 


10.717458 











716775 
716093 
715412 
714732 
714053 
. 713376 
- 712699 
.712023 
10.711348 


Tang. |79° 


LS aaa 





| 





COSINES, TANGENTS, AND COTANGENTS 101 


| 
11°; Sine. ‘| D. 1°. || Cosine, | D..1". Tang. | D.1". | Cotang. 168° 


























0’! 9.280599 | 49 go || 9.991947 42 || 9-288852 | 44 99 | 10.711348 60" 
1 | 1281248 | 19-82 || ‘ogiog2 | <4 280326 | 11-33 | “710674 
2| <esiso7 | 1)-e8 || 901897 a 289090 | 11°59 “710001 3 
Qn yo . 187: ° ; f lod . ; ts 
4 | (283190 0-7 |) coups | 342. || Sogrzae | 11.18 “708058 | 80 
° . é . 3 € . Ory 
5| leases | 45 || .g91e23| 45 || 202013) 37-48 | 707987 | 55 
6 | 2844804 40-4 901799 eS 22682 as T0738 54 
” | (285124 13 |) “99177 295 a "706650 | 53 
8 | 985766 ae 991749 ine "994017 oe "705983 | 52 
9 | (286408 | 49-20 || ‘gov7e4 | “45 odes | 11-12) (705316 | 51 
10 | 1287048 | ip:gy || 991609 | -45 295349 | 11-08) ‘704651 | 50 
11 | 9.287688 | 49 63 || 9.991674 42 || 9.296013 | 44 o, | 10.703987 | 49 
12 | .288326 | 19-68 || “ “oo1649 | <4 206677 | 31-07 | “ “703388 | 48 
13 | 1288064 | 19-63 || ‘oo1e2s | <4 297330 | 11-03 | "702661 | 47 
14 | 1280600 | 39-69 || ‘901599 | 48 298001 | }1-03) "701999 | 46 
Fare bi Saaero | 220.87 |] “gorso | 142 || 2Spason | 11.00 | Sthopers Lag 
7 | ‘991504 Wine "991524 ae "399980 a 700020 | 43 
18 | ‘202137 | 19-55 || ‘ooi498 | -43 300638 | 10-87 | ‘699362 | 42 
19 | (202768 | 19-52 || ‘oo1a73 | 48 301295 | 10-05 | ‘698705 | 41 
20 | ‘203309 | 19-52 || “ogiaas | -42 || ‘301951 | 10-23 | “eggo49 | 40 
21 | 9.294029 ‘o || 9.991422] | 9.302607 10.697398 | 39 
92 | ‘294658 | 19-48 || “‘g91307 | 42 "303261 | 19-90 | ~~ “g96729 | 38 
23 | 295286 | 19-47 || ‘g91379 | —--42 303914 | 10.88 696086 | 37 
24} 295913 | 19-45 || ‘991346 |  -48 "304567 | 19-88 | “695438 | 36 
25 | (296039 | 19-43 || “991391 | ~~ -42 "305218 | 10-85 | — “gga7go | 35 
26} ‘297164 | 19-42 || “ogr995 | — -48 "305869 | 19-85 |  “¢94131 | 34 
97 | ‘297788 | 10-40 || ‘ogia70 | -42 "306519 | 10-88 |  “@g348i | 33 
98 | "298412 aylee "991244 es "307168 ee "692832 | 32 
29 | :290034 | 19-37 || ‘oorgis | -43 || “z0ve16 | 10-8 "692184 | 31 
30 | 1200655 | J)? || ‘991193 | +43 308463 | 19-"8 | 1691587 | 30 


31 | 9.300276 ~ || 9.991167 | 9.309109 10.690891 | 29 
32 | 300895 | 19.32 |] “‘oo1141,| -43 |] {309754 | 19-75 | ~~ “eq0246 | 98 





» 
33 | 301514 toi "991115 es "310899 ae "689601 | 27 
a4 | 1302132 | 39-39 || ‘991000 | -43 || ‘ai1042 | 19-72 | ‘esgo5s | 26 
35 | (302748 | 19-27 || ‘ooio6a | -43 || 1311685 | 40-72 | “egeais | 25 
36 | 1303364) 19-5 || ‘oo103g | 48 || ‘siaae7 | 10-70 |  ‘es7673 | 24 
37 | 303979 | 19-53 || coo1012 | -43 || <312968 | 19-68) — ‘esvose | 93 
as | 1304593 | 10-33 || ‘990086 | 43 || 313608 | 40-67 | “eseag2 | 22 
39 | ‘g0s207 | 13-33 || ‘o90960 | 48 || 314247 | 10-65 | es5753 | 21 
40 | ‘305819 | 70-78 || 990934 | -43 314885 | 10-63) 685115 | 20 


41 | 9.306430 10.18 9.990908 43 9.315523 10.60 10.684477 | 19 














42} 1307041 | 10-38 |) “‘oo0ss2 | -43 || 1316159 | 19-60 |  “ggaeai | 18 
43 | 307650 | 10-15 || ‘oo0g55 | -43 || 316795 | 419-60 | “683905 | 17 
) gee | Bee) oe |) eee] se) ae 
Br 1 areoe 10-20 | ogre | a8 Il Satoaay | 10-581 wrpeyarn | ae 
48 | 1310685 ee "990724 a 319961 ines "80039 | 12 
8) HS or) thier] | Bae) TB) eae | 
50| 311 "99067 "391999 78778 | 10 
10.03 43 10.48 | ° 
Bt | 9.312495 9.990645 9.821851 10.678149 | 9 
10.03 45 10.47 
52 | 813097 | 19°99 || -990618 | “Ze || 322479 | ig-4s | 677521] 8 
Bi | Sistaer | ..8°98 |) Moeases |. 1:42 || taseres |: 10-45 | eterenerl te 
55} 314897 | 29-90 || “ogosag} 49 "304358 | 10-42 |  ‘grse4o | 5 
56 | 1315495 | 997 || ‘990511 | -45 || ‘324998 | 19-42 | “ezs017 | 4 
7 | 1316092 | 9-99 || 2990485 | -48 "305007 | 10-40) “@r4gq3 | 3 
58 | 1316689 9-95 | ‘900458 | “45 || ‘g26081 | 19-40 | ‘erazeo | 2 
Bo | 317284 | 9-92 || ‘oooaai | -42 || ‘azeas3 | 19-87) ‘eraia7 | 4 
60’| 9.317879 -92 || 9°g90404 | 9.327475 10672525 | 07 

















101° Cosine. i D. 1". Sine. Diet Cotang. D1". Tang. 78° 




















102 


TABLE XXV.—LOGARITHMIC SINES 














12° Sine. 
0’| 9.317879 
1 .818473 
2 .319066 
3 .3819658 
4 .3820249 
5 . 3820840 
6 .821480 
cd . 3822019 
8 . 822607 
9 .823194 
10 .3823780 
11 | 9.324366 
12 ~824950 
13 .825534 
14 .826117 
15 . 826700 
16 .827281 
17 . 827862 
18 .328442 
19 . 329021 
20 .3829599 
21 | 9.380176 
22 .8380753 
23 .831829 
24 . 381903 
25 00247 
26 . 883051 
27 . 303624 
28 . 334195 
29 .834767 
30 .8353387 
31 | 9.835906 
32 .386475 
33 .83710438 
34 .3837610 
35 .338176 
36 . 3388742 
ie .383893807 
38 -839871 
39 .340434 
40 . 3840996 
41 | 9.841558 
42 .3842119 
43 .842679 
44 . 848239 
45 . 848797 
46 .3844855 
47 844912 
48 .845469 
49 . 3846024 
50 346579 
51 | 9.347134 
52 . 3847687 
53 . 848240 
54 . 848792 
55 .3849343 
56 . 849893 
57 . 900443 
58 . 800992 
59 .801540 
60’| 9.352088 











102° Cosine. 





Del's 


OD 00 GO GU GD OOS 


woMDOOOOOO 
tI_ 
SBSSSERAZES 


9.77 











Cosine. 


9.990404 


. 990378 
. 990851 
. 990324 
. 990297 
. 990270 
.990248 
.990215 
.990188 
.990161 
.990134 


9.990107 
. 990079 
- 990052 
-990025 
.989997 
989970 
989942 
.989915 
. 989887 
. 989860 


9.989832 
989804 
989777 
.989749 
989721 
.989693 
989665 
.989637 
.989610 
. 989582 


9.989553 
989525 
989497 
.989469 
989441 
. 989413 
989385 
. 989356 
. 989328 
. 989300 


9.989271 
. 989243 
.989214 
989186 
989157 
989128 
.989100 
989071 
. 989042 
.989014 


9. 988985 
988956 
988927 
.988898 
988869 
988840 
988811 
988782 
988753 


9. 988724 


Sine, 








D. 1". 














) 
Tang. | D.1". | Cotang. 167° 





9.827475 10.33 
828715 ; 


329334 ae 
329953 | 10-32 
330570 | 10-28 
: 10.28 


"9331187 
"331803 | 10-27 


"332418 ae 
883033 | 20-25 


.333646 | 39°99 


834871 | 40°18 
335482 
2) 40.18 
336093 
gz6702 | 10-15 
10.15 


339133 
809739 10.08 
9.340344 | 49 07 
840948 | 49‘ o7 
“B41552 








342155 | 10-05 
mee 10.03 
042757 c 
; 10.02 
843358 
2 10.00 
844558 f 
; 9.98 
845157 
345755 | | 8-88 
; 9.97 
9.346353 
‘g4gg49 | -9-98 
Se 9.93 
847545 9.93 
348141 ‘ 
rs 9.90 
349329 9.88 
349922 9.87 
850514 ed 
: 9.87 
.851106 
351697 | 9-88 
: 9.88 
9.902887 | 9.99 
. 352876 ; 
patos 9.82 
3538465 9 80 
.854053 9 "8 
354640 |g "kg 
355227 9.97 
355813 9 
. 856398 9.93 
. 356982 9°43 
oe We 
:357566 9°49 
9.358149 “ 
358731 | 9:72 
859313 9.67 
859893 : 
ia 9.68 
360474. 9°65 
.861053 9.65 
861632 963 
. 362210 9 62 
362787 9.62 
9.363364 ; 


Cotang. D. 1’. 


10.672525 
671905 
-671285 
.670666 
670047 
. 669480 
.668813 
.668197 
- 667582 





. 666967 | 


666354 


10.665741 
.665129 
.664518 
663907 
663298 
- 662689 
. 662081 
.661473 
. 660867 
. 660261 


10. 659656 


10.653647 
653051 
652455 
651859 
.651265 
.650671 
650078 
.649486 
.648894 
. 648303 


10.647713 
647124 
646535 
645947 
.645360 
644775 
.644187 
643602 
648018 
642484 

10.641851 
.641269 
.640687 
.640107 
- 639526 
638947 
638368 
637790 

637213 





Tang. 





| 10. 636636 


SRW WROODIMCO 


~ 





| 
Jj 
oO 


a a EE 


ay 


ENTS 103 
COSINES, TANGENTS, AND COTANG 
























































| 
C tang. 166° 
i Da Tang. D. 1°. | Cotang 
i Dit” Cosine. a be lst 
13°; Sine. Pat's yg A Bhi | 
| a aeeeh 363364 an 10.630636 60 
See ae 7 . Ne. ; 536 
8 9.988724 | 4a 9.363864 9.00 ‘Boa z 
0’| 9.352088 | 9 49 ‘gss6o5 | -48 oe sone | 2 
1| 352635 | 9°49 “GaRBEG 48 seis | saa | & 
ee ee a: ‘res | 38 | B00 ar | Sas | 
Bl iataere |) 9:08 -ose60y | 4g || 365004 | Ga | 
5 | .35481 9.05 ‘gees | -50 so | 3.3 se 
SAE Shor || Noesete a || | Oe soo 8 
ae 0.02 fete i. 38524 9.50 | “C8146 51 
. 5 t ‘ 42 : 
sais 0 oe oes ace aes 10.630837 49 
10 . Yiay zs 9. . : co , 
9.988401 | 54 369663 ‘8 4/8 
11 9. 358064 8.98 cgssari | -50 37082 me so : 
12 epee 8.97 “988342 Es 30709 as sa | 
13 cos 8.95 ‘gpsare | -50 snot) 9% see | 8 
£9 | 800215 |’ 8 'o5 “gssase | -50 ae soa | 
i) se | Sb || ee Bo || -373e29 | 8:45 625807 | 41 
anise 8:90 Hees -50 341983 9.38 | 620807 | 4i 
9 "50 3741 ; 
20 et 8. : . 034 
9.988108 | “55 375819 sr i | & 
= "ok 885 “seus | *80 376442 ae oe 36 
yess | ‘50 || -227008 | 3p 621878 | 34 
24 | 365016 | Q'g5 ‘gsrass | -50 ame) oe sa 
2) ware) 8S ‘opines | 22 -878681 | 9/39 "620761 | 32 
z| some | Se || aes 0 || 379289 | 8-83 620208 | Bi 
ar 88) soto ee S997 12g | 620208 | 84 
284)" ks hr ° "987365 = Pan ie : : 
goats | 8 all oe 10.619090 
30 | 368185 | 3-44 i bee es ora | 2 
— a a "381466 ay BOo4 | 28 
Boas | 8 ‘onriad | <8 -882020 0.05 wees 26 
| cone | 8 “887710 50 -882575 | 993 (616871 | 25 
S| gies | 82 error |. 62 883129 | 9°59 "616318 | 24 
mS | 30a 8.10 “perei9 | -50 -383682 | 9°59 615766 | 23 
ae S10 ‘gpreis | “22 -884234 | 9°59 "615214 | 22 
| aise | be ‘anise | <2 ‘884786 | gig | |: 614663 | 21 
Sin oe ‘gsiser | 22 .885337 | 9°48 "614112 | 20 
a3 | .972373 | 8-35 | ae ae : 
39 piers 8.67 987526 | “59 ress oe 10-1 " 
40 | .873- to ee : | 
9.987496 | “no Os | 9.15 AEE 
a) muse] 86 | ut ‘BE ||) eBstese | 8448 C1196 | 16 
| pee sae mae be || 388084) 9°12 611369 | 15 
B/S ‘anise | 22 -888631 | 9°49 "610822 14 
Drea bp. || ee pe || 888178 | 940 610276 | 13 
a) aie 50 ‘oniao | “32 889724 | 9/19 609730 | 12 
| “sro | $90 | Poh te 890270 | 9'o8 "609185 | 11 
48 | 3rinag | 8-51 987279 | “59 .390815 | 99g "608640 | 10 
ag) seas | 8 comet) ae 203860 |!) Son| |b 
a) at) 33 | sean . | 10.608097 9 
50 | .87877 | 8-32 teen i 2 gory on a 
Lad re : : i 
Ba | “sraeol | 833 eres st abe 392989 | 903 ye 6 
m3 | ‘sou Spe || -9eraes 58 . 393531 9.03 605927 | 5 
ere cot | eae ‘52 | -B!073 | 9 'p 605386 | 4 
ene | costoo | 8 894614 | 909 "604846 | 3 
ate sab. | aoe Pe -895154 | 9/99 “604306 | 2 
Br | amie | S48 ee | eR .395694 | g'9g 608767 | 1 
| iene | 6 | copeage | 2 -896233 | 8" 97 10.603229 | 0 
58 .382661 8.45 | 986936 “33 eee : io 
0’! o’aeaers. | 8-45 || 9‘986904 ee 
en ~ Si D. 1". || Cotang. | D. 1’. 
103 i D. 1". || Sine. - aD: 
103° Cosine. | D. 1’. 








(SS aS ES ES SE TD 


104 TABLE XXV.—LOGARITHMIC SINES 





| 
14° Sine. D. 1”. || Cosine. | D. 1". Tang. D. 1". | Cotang. 165° 












































0’ 9.288675 8.45 es se || 9- tte) 8.97 Bec 60’ 
1 | .384182 ; . 98687: ; 5 , .602691 | 59 
2| 1384687 eves "986841 | - = "307846 ape 602154 | 58 
3} .885192 | 3°49 . 986809 52 .398383 | gos 601617 | 57 
4| .385697 | 3°49 .9867'78 33 398919 | 8'93 601081 | 56 
5 | .386201 | 3'3g . 986746 53 399455 | 3°99 600545 | 55 
6 | .386704 | 9'3e .986714 50 -899990 | 8°99 600010 | 54 
7 | .887207 | g"3e . 986683 5a 400524 | 8°99 599476 | 53 
8 | .387709 | B55 986651 "53 .401058 | 393 598942 | 5 
9} .388210 | p35 .986619 53 401591 | 9 "pe 598409 | 51 
10 | .388711 | g'33 . 986587 53 402124 | 3gr .597876 | 50 
. 11 | 9.389211 |g 99 || 9.986555 53 || 9-402656 | g gx | 10.597344 | 49 
12 | 889711 | 3°39 .986523 53 -403187 | g°o2 .596813 | 48 
13 | .390210 | 9°35 986491 3 .408718 | g°ge .596282 | 47 
14 | .890708 | 6°39 986459 53 .404249 | 8° G0 595751 | 46 
15 | .891206 | "53 986427 33 404778 | B°o5 595222 | 45 
16 | .391703 | 9°S- 986395 33 .405308 | 3° g9 .594692 | 44 
17 | .392199 8.0% . 986363 33 -405836 | 929 594164 | 43 
18 | .392695 8.97 986331 3 .406364 | 9°a9 593636 | 42 
19} .898191 | 65s .986299 “BR -406892 | png 593108 | 44 
20 | .393685 | 9°53 . 986266 “33 407419 | Bae 592581 | 40 
Bf | 0H gine [SRR con || CARREY |nniee be 
23 | .395166 | B°59 || -986169 | “53 |} .408996 | Bins .591004 | 37 
24] 395658 | 9°55 986137 “55 409521 8.43 590479 | 36 
25 | .396150 | 94g . 986104 53 .410045 | "es 589955 | 35 
26 | .396641 | 3°48 . 986072 55 .410569 | g°45 .589431 | 34 
27) 897182) gis .986039 “33 .411092 | 9°05 .588908 | 33 
28 | .307621 | B42 -986007 Br .411615 | 3'np 588385 | 32 
29 | .808111 | gy. || .985974 | “po || 412187 | B°63 .587863 | 31 
30 | .398600 | 9°33 . 985942 35 412658 | 8" 6g .587342 | 30 
31 | 9.399088 9.985909 9.413179 10.586821 | 29 
32 | 1399575 heed 985876 | “> |] 1413699 aes "586301 | 28 
33 | .400062 | 9°45 .985843 53 .414219 | 3° px 585781 | 27 
84] 400549 | 3°35 985811 oe 414738 | 8's 585262 | 96 
35 | 401035 | gg 985778 “35 .415257 | B69 584743 | 95 
36 | 401520] gg 985745 35 415775 | 9’ ¢3 584225 | 94 
37 | .402005 | gos 985712 “55 -416293 | 860 583707 | 23 
38 | .402489 | 9px .985679 BB -416810 | 9°69 .583190 | 29 
39 | .402972 | Bos . 985646 Fe 417326 | 3°60 582674 | 91 
40 | .403455 | 99s . 985613 35 417842 | 3°60 582158 | 20 
41 | 9.403938 | 9 3 || 9.985580 55 || 9-418858 | g xg | 10.581642 | 19 
42] 404420 | o'pp 985547 pd 418873 | 9 "5° .581127 | 18 
43) .404901 | 9'5 985514 £7 419387 | "56 .580613 | 17 
44| .405382 | 9'o9 .985480 85 -419901 | "55 .580099 | 16 
45 | .405862 | "9g 985447 “5B -420415 | B"p6 579585 | 15 
46 | .406341 | 59g 985414 BE 420927 | 95s .579073 | 14 
47 | .406820 | og 985381 “BP 421440 | "59 .578560 | 13.” 
48 | .407209 | ogo 985347 oe 421952 | "55 .578048 | 12 
49} 407777 | 79x 985314 3 422463 | 8°55 577587 | 11 - 
50 | .408254 | op . 985280 Bs 422974 | 3°25 .577026 | 10 
51 | 9.408731 9.985247 5 9.423484 10.576516 | 9 
52 | 409207 Ui 985213 | “pe || 1428993 =: = "576007 | 8 
‘53. | .409682 #99 . 985180 Br 424503 | 3° 4 .575497 | 7 
B4 | 410157 | 95 . 985146 35 425011 | 3 4o 574989 | 6 
55 | .410632 790 .985113 7 .425519 | 3" 47 574481 | 5 
56 | 411106 | 4°53 .985079 “By 426027 | 6 "4s .573973 | 4 
aemealto,9 | tm 88 985045 a7 426534 | 9" 45 .573466 | 3 
58] .412052 | 45 .985011 “55 A21041 | 9°43 572959 | 2 
59 | .412524 787 . 984978 a 427547 | 8°45 572453 | 1 
60’ | 9.412996 : 9.984944 : 9.428052 : 10.571948 | of 
—| —__ a - ee | ens 
104° Cosine. | D.1". || Sine. | D. 1". |! Cotang.| D.1..| Tang. |75° 


ae rrr nt rr a SE RRS ein a ES a ee 
© v) - 
t 


COSINES, TANGENTS, AND COTANGENTS 


15°| Sine. >, i 


0’| 9.412996 











1| .413467 Bee 
2} .413938 | 2°93 
31 .414408 783 
4| .414878 789 
5 | .415347 780 
6| .415815 | f 
Pr a ay) « » % . 80 
Z| 416283 |" 5 "E0 
Be 141671 | SS 
9] 417217 | ote 
10 417684 ae 
11 | 9.418150 | .», 
12 | .418615 m9 
13} .419079 ws 
14 | 419544 | Aof5 
5) 480007 1) eS 
-16 | .420470 ae 
7 | (4999083 | &-¢ 
18 | .421395 | 4: 
19 | .421857 | +63 
20 | .422318 oer 
21 | 9.42277 
99 | .493238 Gee 
23 | .423697 7 65 
24 | .424156 | 4-32 
25 | 424615 | 4°63 
26 | .425073 69 
27 | .425530 762. 
28 425987 | * 85 
29 | .426443 | 2°65 
30 | .426899 ] f56 
31 | 9.427354 
32 | .427809 | 7-58 
83 | 428263 | fps 
34 | 428717 | oe 
85| .429170 Be 
86 | .429623 a 
87 | 430075 | abe 
88, .480527 | «425 
39 | .430978 | A55 
40 | .431429| 4°25 
41 | 9.431879 | . .4 
42 | .432829 | 2-43 
43 | 482778 | aie 
44) .433226 | 278 
45 | .433675 45 
46 | .434122 oe 
47 | .434569 | 4-7 
48 435016 |» 3 
49 | 435462 | 2-73 
Hee 
< Ie 
52 | .436798 on 
53 437242 | 2°75 
55 | .438129 | +38 
56 | .488572 | "ge 
57 | .439014 | 5, 
58 | .489456 : 


7.85 
59 | |439897 
60’, 9.440338 7,85 


105° Cosine. | D. 1’. 

















Cosine. | D. 1". 
9.984944 
"984910 Bs 
98487 a 
984842 
984774 a 
984740 Be 
ass06 i 
98467 
984638 | 34 
984603 | -P8 
9.984569 os 
984535 | “Pe 
984500 | 28 
984466 | “pe 
984432 58 
1984397 | -3 
984363 | “4 
"984328 | -28 
984294 58 
984259 | ->3 
9.984224 2 
984190 | “pi 
984155 | “PS 
984120 | “28 
984085 58 
984050 | “Ps 
984015 | “PS 
983946 | “38 
oe os 
983875 
983840 "es 
"983805 | “33 
983770 | ‘Pg 
1983735 | “Pe 
983700 | “25 
983664 | “$8 
983629 | "D3 
983594 | “25 
a = 
98352: 
“983487 iG 
983416 | °2 
983381 | *33 
983345 | 60 
983309 | “po 
983273 | “Pp 
983238 | °26 
Te i 
9831 
"983130 a, 
983094 | “By 
983058 |. “0 
983022 |  °f0 
982986 | “By 
982950 | “8 
og2014 | 6° 
98287; ae 
9.982842 | - 
Sine. Deis. 








Tang, 


9.428052 
428558 
429062 
429566 
430070 
.480573 
431075 
431577 
-482079 
482580 
.433080 


9.433580 
-484080 
.434579 
.435078 
-435576 
.486073 
436570 
437067 
437563 
.438059 


9.438554 
489048 
439543 
.440086 
.440529 
.441022 
.441514 
-442006 
442497 
.442988 


9.443479 
.443968 
.444458 





-444947 
445485 
-445923 
.446411 
.446898 
447384. 
447870 
9.448356 
-448841 
449826 


-449810 * 


450294 
450777 
-451260 
451748 
452225 
452706 


9.453187 
453668 
.454148 
454628 
.455107 
455586 
-456064 
-456542 
.457019 

9.457496 





_Cotang. 








D. 1’. 


ost 


SE2S2SESQC90 Be EB EPH YH HBR He HHH WON NNNWD WW 
WW WOLOAUDTNINDD GDOOW WWW DHDDMDOONWWW OC 


[SE fe 
CXS) 


NojvezaaKa) 
eo eke oa) 





) Tang. 


105 


| 
Cotang. 164° 


10.571948 
571442 
.570938 
570484 
.569930 
569427 
568925 
568423 
.567921 
567420 
566920 


10.566420 
-565920 
.565421 
964922 
564424 
563927 
563480 
562933 
562437 
.561941 


10.561446 
-560952 
560457 
559964 
059471 
.558978 
.558486 
.557994 
.557508 

557012 


10.556521 
556032 
555542 
.555053 
554565 
554077 


-553589 | 


.553102 
.562616 
-552130 


10.551644 
991159 
1550674 
.900190 
.549706 
549228 
548740 
548257 
5477S 
547294 

10.546813 
-546332 
545852 
545872 
.544893 
544414 
543936 
548458 
542981 

10.542504 





-|— 


NH, 





— 


16°; Sine D 
& a Cosine i 
eet Tang. | D. 1". | Cotang 163° 

































































0’ | 9.440338 | oe a 
440888 7.38 9.962842 62 || 9.457496 Lip |e 
"441 na) || cbeean | . 
3 Sete 733 “ogo769 |  -60 457973 | #93 53027 | 59 
4| 442096 | @-30 982733 | -B eee |e ae: 
5 | (442535 | 0-82 982696 | * 55 S| oe Sa | 
6 | ‘442973 | 7-30 | "982660 |  -89 59400 | gp Bion 36 
| dee is inn | 450400 | 799 "540600 | 56 
g| _4a43e47 | 2-38 982587 | “63 “160823 a Sal 
9 | ‘444984 | 7-28 982551 | -60 “01297 70 sie 8 
OF 1 Wopiareg |) Get. ||. Weeeees | ae | a 
1 7.25 “ggoayy | 82 AOL | 89 B30 5 
oe 2 461770 | 757 “538230 51 
13 | l4dgoes | 223 || ‘gsesa7 2 | toe E 
14 | 1440459 | 2-28 982367 -62 43186 | 8p Sais 48 
15 | 1446803 | 2-23 "9g2331 |  -80 463608 | 1s ‘a0 ci 
16 | 1447306 | £22 - 982294 ia Bae | Ess on 
17. | ‘aav759 | 2-88 "ggo057 | 82 461509 | 73 Biol 45 
18 | 448191 | 2-20 “gg2220 | 88 465009 | F's3 Sut 4 
49 | 449603 | 7-20 “gg2183 | -82 465589 | 7 : i 8 
a9 | 1449054 | 4-18 “ogoi46 | 82 “466008 | 4°55 “pase 42 
: rs St | 8 -466008 | 7's "533992 | 42 
1 a i 466477 | 89 "533523 | 41 
bo | -aagors | 2-22 || 9 Geposs i | re | aR 
| tous | 2H 989085 “60 467880 0.78 rae oo 
Be | Pegi || GelBy || eet ing 62 |) insets | 2 oe 
96 | 1451632 | 6-23 .981924 oe | ae sas : 
97 | ‘452060 | 2-18 981886 | °63 “40976 cn 8 : 
ge yaniosies |) Da. | Wontar isa || 400186 | pots pio 
Beaters | ote. Meters ee || a | 2 ee 
Mee eis | Cae | Aaa em ce | at 
a fina | 410678 | 7.75 "529324 | 32 
31 | 9.453768 | » ey ae Sat | 
8 | oaeuio4 | 710 |l Getbo2 wor 1 ors. 
33 | cai19 | (08 || “getees | ee] FR |” 
peas | COR Nee |) ee | ce oe | & 
35 | 1455469 | 7-08 -981587 oa Payee ‘0 Pans z 
3¢ | ‘455803 | ©-07 981549 | $3 “413910 ma a = 
33 | "456739 | £:05 “981474 -63 tier ve soon) 2 
| tas |r gut | 3 “82 7.68 “520619 24 
40 | 457584 | 6-98 981399 | “62 “afi br ait ® 
41 £08. iL MRgetser fr BS 1 | oe | Sau a 
te 63 AISTOS | 7 Gy "524237 | 21 
| suas | 2° ote i 6 "523777 | 20 
43 | 1458848 | 2:9 981285 | 63 pit : 
ie Aa ges | 63 “Agis2 7.65 523317 19 
46 | .460108 | ¢-9 -981171 ae S517 a Sa i 
47 | 1460527 | 8-98 981188 | 0°83 “T8075 163 ae 
Te areas. |i Gs0h || Bk aetna | ANS | Fe ae 
49 | 1461364 | ©-9¢ 981057 .63 479432 one 20568 13 
Be) PeGprecs |; Gee |] Reena ar ae trope, [Aa 
5 6.97 || “Sengst | -63 || “soso eo | BN 
1 | 9.462199 _ eo | 0 sini 1 
be | 462616 | 6:95 || °-Sso904 rie [OEE 9 
erste: 980042 | 63 9.481257 | « 10.5187, 
56 | .464279 | 8-92 980789 | “62 ore | re sie 3 
Bo] Saaneees | 6,02, || teagee | ite | ta ee a 
| Biol ee ‘ose 8 4395099 | 0-37 516925 5 
59 | 1465522 | 8-90 980673 | °f3 ae | ate ut 
60’! 9.465935 | -88 980635 | "ge ee | oe iss | 
ri Si see o'beosee | <6 || of4sessa 788 aie | i 
106° Cosine 1" so sd 
.| D1" || Sine. | Di 1 iets aed 
i 174 |} Cotanze | Dare 713° 
wae. Tang. \73° 








» 


COSINES, TANGENTS, AND COTANGENTS 107 







































































| 
17°| Sine. D. 1". || Cosine. | D. 1’. Tang. | D.1". | Cotang. 162° 
0”| 9.465935 |g ge | 9.980596 63. || 9485389 | 5 5 | 10.514661 | 60’ 
1 | 1466348 | 8-88 990558 | “G2 || 485701 | 428 ‘514209 | 59 
By ceaerizs, | Peete? || ocoaay | 205 || Ssapaggs | 7-820 || setae 18 
a “asane | Ot "980442 | °83 || “4grt43 | 7.50 "512857 | 56 
ol ihre As "930403 | — -89 “487593 | ~-50 512407 | BB 
BA ccapeac? ye, O18 “9g0364 |  -69 "488043 | 7-50 511957 | 54 
v | ‘aegsi7 | 8-83 "980325 | 89 “ggs4g2 | ©-48 "511508 | 53 
G | desea | 6:88 “980286 | - ‘4gso41 | /-48 "511059 | 52 
"469637 | 8-83 ‘ggo247 | “8° "4gg300 | © -48 "510610 | 51 
Be IRE ceed | OD Ah gi wae |) Teta 
10} ‘4700s | 8-8 980208 | “62 40838 | 1-47 "510162 | 50 
11 | 9.470455 | 4 g4 || 9.980169 gs || 9-490286 | » yx | 10.509714 | 49 
12 | .470863 | 680 || “‘ogoiz0 | —-85 490733 | 4°43 "509267 | 48 
B) dae ss | Rae] | Rae Te | ae 
15| ‘arz0se | 678 || ‘ogooiz | 8% || ‘ag2073 | 7-43 | “Borger 45 
16 | la72i92 | 6-27 || ‘gv9973 | -85 || “4g9519 | 7-43 "507481 | 44 
17 | lazagos | 8.77 “grgg34 | -O3 "492965 | @-48 "507035 | 43 
48! 473304. | 8-7 “gvoso5 | -83 ‘493410 | 7-42 "506590 | 42 
19 | cavario | ©-% || “gropes | -8% |] “agseea | 7-40 | -Poeiae | at 
19 | 478710 | 6’¢5 Jaleo 65 carpetsteea mT SD: ee 
20 | 1474115 | 6-8 "979816 - 494209 | 1-48 "505701 | 40 
21 | 9.474519 | 6 ag || 9.979776 gs || 9-494743 | 99 | 10.505%57 | 39 
tiie | om | ae | | ae | 28 | Mia 
et cen | 822 "979658 | -8 496073 | 2-38 "503027 | 36) 
Bees | SUR Tl <geria |i 262 496515 | 34 503485 | 35 
Bee, | (8 soyany9 | 162 “496057 | @-3¢ 503043 | 34 
a7 | caveoss | 6-70 || “orossg | -87 || “4oraoa | 7-37 | -Booeot | 33 
| arma | 67 || -grogoa | -8% || “aoren | 7-87 "502159 32 
ge ORS | terre 1) 007 lll qetgens 1) 080" 1) ntepeeeg te 
29 | 477741 | ggg || 9794 "es || -498282 | 7°33 501718 | 
30 | .47ei4e | 8-88 "979420 2 498722 | 1-33 "501278 | 20 
31 | 9.478542 |, ey || 9.979880 e7 || 9-499163 | » 33 | 10.500837 | 29 
2 | argos | 8-87 979340 | -67 || ‘499603 | 7-83 “500897 28 
ey | 88681): conn 1 18% ll mepment | Tee |) anes ae 
Be tena) | 088" ‘g7gen0 | «8% "500920 | #-82 "499080 | 25 
Rl gana | 6:85 ‘979180 | -8% "501359 | 2°22 "498641 24 
oy) faces, | 8782 ‘979140 | -82 501797 | 7-80 "498203 23 
87 | .480087 | . 6 ’g9 97 67 S019" | 7/39 -498203 | 2 
38 | 4gias4 | 6-62 979100 a “502235 | 1-30 497765 | 2 
39 | 481731 | 6-68 979059 | -88 “502672 | 1-38 1497328 | 2 
40 | laszies | 6-68 979019 ev || sspoatog | 2-38 "496891 20 
41 | 9.482525 | @ gq || 9.978979 g7 || 9-30846 | » oy | 10.496454 | 19 
de | qdazaoi | 6-00 gis | «68 503082 | 7-30 496018 | 18 
5 DEN . lod . : Or DD | 
BASS) So) | See) | aM) Fa | dae 
5} \44i07 | 6-58 |) ‘oregiy | 68 || “xosegg | @-25 "494711 | 15 
45 | asasoi | ©-8¢ || cotarry | <8? || ‘Posroa) 7-25 | “douv6 | 4 
“484995 | 8-57 ‘grsvs7 | -6% "306159 | 2-25 “493841 | 13 
as | Asemag | 8-57 "973696 | -68 506593 | 6-23 "493407 | 12 
re 5682 | 8-55 || ‘ores | -68 || ‘sor0a7 | 7-23 | ‘490073 | 11 
49 | .485082 | 655 “978051 ‘67 507027 | "93 A929 
50 | <486075 | 6-22 || 978615 je “507460 | 4-33 "492540 | 10 
BI | 9.486467 | ¢ xx || 9.978574 63 || 9-307893 | » 99 | 10.492107 | 9 
Be | 486860} 6:73 || ‘ovesaa | -68 || “‘sos326 | 7 22 491674 | 8 
S| caernas | (S88 8ll porwane } 88° ll) Ceopter.(/ 222° | iaooeen | 
ee. | | ete “‘ovsati | -88 "509622 | 4-18 "490378 | 5 
55 |  .488 65a. || -2eeat "68 -509622 | 7/90 49034 
b6 | 1488424 | 6-50.]; ‘oveszo | 68 510054 | 4-22 489946 | 4 
py | .4sssi4 | 8-30 o7es20 | -68 510485 | 4-38 "489515 | 3 
58 | .4sg004 | 6-20 || .overas| 68 || “siogie | 7-38 "499084 | 2 
59 | 1489593 | 8-48 orseq7 | °68 511346 | 2-17 "488654 | 1 
60’| 9.489982 | -°- 9.978206 | - 9.511776 | * 10.488224 | OF 
107° Cosine. | D 1". || Sine. D.1". || Cotang. | D. 1’ Tang. |72° 


108 
TABLE XXV.—LOGARITHMIC SINES 



























































18°} Sine. Dit? Cosi 
ig) Et sinew s. 3 | 
i is 1 Tang. | D.1". | Cotang. 161° 
0’| 9.489982 & Eo 
982 | 5 4g || 9-978206 . 
Bp aegeca | Gra saree te C68! |) eens en | eco 
2| 400759 | g-4y || -978124 268) (1 Se aeeenes ves 487794 | 59 
3) dud | 647 Pane |e SEBO HT ge eee ee .487365 | 58 
4 491535 | 8-47 || ‘orgoaz | -68 “513064 | "45 -486936 | 57 
5) got | 5-43 || 98001 "68. || 218493 | 743 :486507 | 56 
: 492308 | 6745 || -977959 stmlbee reed wars -486079 | 55 
q 492695 | 5-43 || ‘orzoig | 68 || ey eka ET “485651 | 54 
8) 493081) G49 || 27a’ 68h |) eer Tee 485223 | 53 
10 | <ao3eo1 | ©-42 |] “S7yrod 6g || 313681 | 35 “484369 | bi 
alt ea ee 7 sien?’ | 710) ssanagae | oy 
NOES Fyre RL Oba Dinar che gorau |! | spas pe 
2 -aodizt | 6-49 || -geazi4 68 || 9: » 19 | 10.488516 | 49 
13 | 495005 | 6°38 || 977069 70 |} MORO.) 7cuga | eerie 
ue pics 88 | 6°40 97°7628 .68 c Bien 7 10 .482665 | 47 . 
as (72 | gon 977586 10 SITT61- |e” 08 .482239 | 46 
6 | 406154 | Frse || 97744 | 3 -518186 | 97 481814 | 45 
17 | 496587 | g'37 || 977508 "68 || -218610 | v7 -481390 | 44 
18 | 496919 | gay || 97461 "70 || -219084 | a o7 .480966 | 43 
20 cagvosz | 635 || ‘orese7 0 spigase | 0 | Gnotis | 41 
: 97937 ni : " 4801 
oi | 9.49s0e4 | os eA el bie 108 “479605 | 40 
22 | .4984 6.83. || 9-277835 ra’ || 9.5207 : 
an 44| Oras “‘ovyo93 | <0 -520728 | wy gp | 10.479272 | 39 
7 : 498525 6. 3 9 977251 a7 52 M151 708 -478849 | 38 
24) 49904 | G33 || 977209 |} B21573 | 7'g3 |  -478427 | 37 
25 | 499584 | B39 || 977167 "wg || -521995 | 2°93 | —-478005 | 36 
26 | .499963 | 6-35 || lorries | -f -BR2417 | "9 477588 | 35 
Bi | D002 | g's || 277088 "70 || -522838') 709 477162 | 34 
28 | 500721) G39 || 977041 "70 |} -328259 | 799 476741 | 38 
29 |. :501099 | 8-89 || ‘oveo99 | «70 523680 | £95 | 1476320 | 32 
30 | -501476 | 6-25 oveony | her » 00 475900 | 31 
2 ~ Z . w72 -524520 ; 4 
a, oamsst| ay || oatsent| am || 9.92000 | a | 1.ezm0 | 2 
33 | 52607 6.20" || garoean |: 220 525359 | Beg “dred | 33 
a ’ Sie 697 976787 ave : ore : 8 6.98 .474222 | 27 
B> | 503360 | G55 || 916745 "70 || 226197 | 97 :473803 | 26 
= Pi be 6 05 976702 2 seein 697 .473385 | 25 
; . aa 6 95 976660 70 : 24033 6.97 472967 | 24 
Bs | 50488 | GOS || coro6rT | eee | 66th | eee 
BH 5046 693 976574 Pat ares 6.95 .472132 | 22 
505234 976 70 || -528285 7 
6.23 || 976582] “bes7os | 8-25 |  “grigg8'| Sp 
41 | 9.505608 % 02 6.95 -471298 | 20 
42 505981 6.22 9.976489 ne 9.529119 : 
aa eee. 6.99 976446 Se 52 6.98 10.470881 | 19 
oF 35] 6354 629 976404 70 -529535 6.93 .470465 | 18 
vie . us (27 620 976361 Ais 5 529951 6.92 .470049 | 17 
45 | 501099 | ¢°59 || -976318 "72, || -330866 | 99 469634 | 16 
6| :50r471 | 6-20 || ‘ova | -% 530781 | g’g9 |  -469219 | 15 
41 BOMB) Gig |) 976232 72 -531196 | "99 .468804 | 14 - 
48} posit | gig || 976180 2: -531611 | 699 "468389 | 13 
‘sosses | 8-38 || “orera6 | -2 532025 | Bon | 467975 | 12 
50 | .508956 | 2° a 7m ||. 082489 | 6” 467 
6.17 || 976103 | on “bases | 6-90)  “genay | ag 
51 | 9.509826 | oa || 9.976 $ 6.88.5) » “Oa 
Dit epee BAT Tl sees no || 9-588266.| ¢ 
615 976017 ai 53367 6.88 10.466734 | 9. 
53 | 5100865 | gig || -975974 ab ene eS -466321 | 8 
4 | 510884) G45 || 975980 "73, (|| -534092 | a7 -465908 | 7 
Beenie | ie ib oll ae a, ||: RE 6 eral pee 
Bee ee nee? | tiga oll (ees Be |) eres | Oe rae 
Bh Sea | 1648.4 ae 73 |) “paena9 | 6.85 | “4eanei | 3 
eek ees, | C18 dices 72 |] -B5eie9 | 6-85 | “Zeger | 3 
Behn | (6.19 Wl ee a 536 6.85 -463850 | 2 
60" | 9.512642 | beercalc: 7. {| igteennea geile Be Sar ceea seal aae 
108° Cosine. | D. 1". Sine. | D.1” a rsa 
. . 1",)}| ‘Cotang. | D.1" Tang. (|71° 








COSINES, TANGENTS, AND COTANGENTS 


109 





19° 


10 


59 
60’ 


DPHIAMABWVWES | 





Sine. 


9.512642 


518009 
513375 
.513741 
.514107 
.514472 
514837 
.515202 
515566 
.515930 
516294 


9.516657 
517020 
517382 
517745 
.518107 
.518468 
.518829 
.519190 
.519551 
.519911 


9.520271 
520631 
520990 
-521349 
521707 
-522066 
522424 
522781 
523138 
523495 


9.523852 
524208 
524564 
524920 
525275 
525630 
525984 
526339 
526693 
527046 

9.527400 
527753 
528105 
528458 
528810 
529161 
529513 
529864 
-530215 
580565 


9.530915 
531265 
581614 
531963 
.532312 
532661 
533009 
533357 
583704 
9.534052 








19) ee 


109° Cosine. | D. 1", 


ee a ntl 











c Relive} WDOODOOOONO Sesosoeoes soeos 
92 90 3p 9p 0p 98 90 0B 90 90 90 90 00 GBH GP OHS PCO LCSLS KKKKSESERS SSESESSRSR KSQLSE 








Cosine.. 





9.975670 
975627 
975583 
975539 
975496 
975452 
. 975408 
.975365 
975821 
975277 
975233 

9.975189 
975145 
.975101 
975057 
.975013 
. 974969 
- 974925 
. 974880 
974836 
974792 


9.974748 
974703 
974659 
974614 
.974570 
974525 
974481 
974436 
974891 
974347 

9.974302 
974257 
974212 
974167 
974122 
974077 
. 974032 
973987 
973942 
. 973897 


9.973852 
973807 
973761 
973716 
973671 

» 973625 
. 973580 
9735385 
973489 
-973444 


9.973398 
978852 
. 973307 
978261 
978215 
.973169 
-973124 
973078 
973082 

9.972986 





Sine. 











DF 1": Tang. 
mg || 9-536972 
73 .537382 
ee 537792 
19 538202 
“3 538611 
“3 .589020 
"9 539429 
“9 .539837 
"9 540245 
“3 .540653 
73 .541061 
wg || 9.541468 
“m3 541875 
“ng 542281 
3 542688 
aes 543094 
WS 543499 
ie 543905 
“an 544310 
“ne 544715 
“mg 545119 
ms || 9.545524 
ce 545928 
ae: 546331 
wg 546735 
fe, 547138 
‘n9 547540 
ns 547943 
“hs 548345 
“ee 548747 
ws 549149 
we || 9.549550 
“We 549951 
“es ||. 550852 
ws 550752 
“ws 551153 
“hs 551552 
“ne 551952 
“is 552351 
“ae .552750 

BD: 
ae 558946 
“a 554344 
“ne BBAT41 
ne .555139 
“oe .555536 
“ne 555933 
“nn 556329 
“hs 556725 
net hee 
5BT51 
TT |) 1557918 
ne .558308 
on 558703 
“hn 559097 
“we 559491 
“an .559885 
tr 560279 
ay 560673 
: 9.561066 


DAS “Cotang. 





Dis 


=~ @ QO GO GO GO GO GO 


aI~3 
IO 


~2 


6.7 





6.60 


Or oror or OF OF OV OT SD PORES 
| ARNRINININIDWOSOS OS 


| cocascace ose 


DA" 


| 
Cotang. 160° 


10.463028 


-458939 


10. 458532 
458125 
457719 
.457312 
.456906 
.456501 
-456095 
455690 
455285 
-454881 


10454476 
.454072 
.453669 
-453265 
452862 
-452460 
-452057 
.451655 
-451258 
450851 


10.450450 


446851 


10.446452 
446054 
.445656 
.445259 
.444861 
.444464 
.444067 
.443671 
443275 
442879 

10. 442483 
442087 
.441692 
.441297 
440903 
440509 
.440115 
.439721 
.439327 

10.438934 





Tang. 











J 




































































| SESSERSRES 





a 
A) 
_° 


110 TABLE XXV.—LOGARITHMIC SINES 
. ” . | 
20°; Sine. D. 1". || Cosine. | D. 1". Tang. | D.1". | Cotang. 159° 
0’) 9.534052 eo || 9.972986 am || 9.561066 : 
1| 534399 | 6-78 || coreo4o | -#7 || 61459 | 3 1° 38541 
| isaqzas | 5-7 || loresoa| 1% || lpeie5i | S22 | lasso 
3 | [535092 | 5-78 972848 | oe 562244 | 8-55 “4377 
4| 585488 | 5-7 ‘o72802 | he 562636 | 8-23 ee 
5 | 1535783 | 2-45 ‘o7o755 | 8 563028 | 8-53 ores 
6| 536129 | 3-27 ‘o72709 | ae ‘563419 | 6-52 it 
7 | lp3ed74 | 2-05 "972663 | °64 "563811 | 8-53 ea 
g |. cages | 23 || iomeiz | f2 || bedpoa | 828 “435798 
9) 537163 | 3-28 972570 | “it 564593 | 6:08 435407 
10} 537507 | 5-43 || fovea | -F7 || 864988 | G59 "435017 
11 | 9.537851 9.972478 9.565873 | 0.43: 
12 | 538104 | 5-22 |] 972431 ze Besran | Gcb0 | Oat 
13| l5g8538 | 5-4 ‘g72ae5 | «it "566153 | 8-50 338. 
"433847 
14| 538880 | 2-00 g72338 | ° 48 566542 | 9-48 
15 | 1539293 | 9-4 ‘g7ee01 | +8 "566932 | 9-50 pee 
16| ‘530565 | 5-02 || ‘g7ao45 | 7 || “xerg20 | 9-4¢ eon 
17| 539907 | 2-4 ‘or2i98 | 48 567709 | 6.48 eee 
18| ceaoag | 5-68 || ‘ori | 8 || “exoos | 648 | aato08 
19 | 1540590 | 9-68 ‘972105 | he "568486 | 8-40 pricy 
20 | ‘540931 | 8-68 |) ‘972058 08 ll copbenera Otay ere 
; All = 
a1 | 9.541272 9.972011 9.569261 | a 0.4807 
e2 .5u1613 | Bf | 71964 18 || “i56068 | 8-43 430852 
a1} “sineos | 5-87 |] “ortsro| 78 | “prose | 649 | Yapogze 
B5 | 512032 5.65 |) ‘ovises| -8 || ‘eroeog | 6-48 “20101 
6 | 549071 | 5: ‘o71776 | 571195 | ° "42880 
or | ‘assio | 5-65 || correo | 78 | ioriset | 843 | “aositg 
25 | “busca | 5-6 || ‘orose| <8 | ‘ortoer | 643 | “demos 
29 | ‘o1sasr | 5-03 || ‘ortoss | 78 |) cores | 642 | “dorods 
O07 | 5-68 a 78 polka -427648 
3 — pile ainiOes 48 || 372738 ans "427262 
32 | 545000 | 3-62 on ee oor (ee pr ise 
ek lbeaepat 6: i 7 ah 6.42 42 
S| EOS Fc eG | Seneeeee Pango itt BBO A egal tee 
7545674 | 2°60 || cov1a9 BO || .Br4e76 "425724 
35 | 1546011 “971851 | 574660 | &-40 , 
36 | 1546347 | 2-60 971303 | “82 || 575044 Hi se 
37 | .546688 | 2: -g71056.| <f 1575427 | 2: “42457 
33 | ‘517019 | 5-6 || ‘griaos | 80 || “pesto | 8-88 | “aoti90 
39 | iota | 5-58 || ‘ortiet | <7 || ‘areios | 888 | “aoanoy 
cri ears FeaebS || Meaeeaes | 200 JU eee, OMe eee 
ok eeaent cle nee 3 ae -Di0DA6 on "423494 
42 | 548850 | 8-08 “HOB = "pret | 6-82 | Ot oo59 
43 | 549603 | 9. covog70 | °89 || carvan |) 68h | ae: 
41 | ioagoz7 | 5-87 || coro | 82 || ereiog | 9-3 | “apteon 
45 | 2549360 | 5-55 || ‘grasa | -82 |) prease | 88% | aorta 
46 | io49603 | 5-55 | ‘oreer | 78 |) presor | 8-85 | “aptiga 
47| “550026 | 5-5 || corm | 82 |) staoss | 8-85 | “apozse 
Fad detec nsisa, || CMe mere 80 || See TaeeS: | eaeaBy. |) ares owe 
49 | Bonde mee “tOS3 a4 "580009 = “419001 
551024 | 2:2 970635 : 8 “4196 
52} 1551687 ee shag 4 ee poe ene 
53] .552018 | 2-2 970490 | * 581528 | 8-2 “41847 
54 552849 5.52 ry .80 “4 6.382 -418472 
55 | 552680 | 5-52 neem -80 pred 6.32 ame 
56} 1553010) P-2? || ‘ovoais | -8% || c5ez6es | 6-85 aan 
57 | 1553341 | P-32 || 970207 -80 || T5e3044 | 8-32 "416956 
58 | 1553670 | 2: ‘g7o249 | £9 583422 | 6-30 4165" 
59 | “554000 Bente’ protease |) > oe || 2s6e9800 | oie, eae 
go" | 9.554820 |? 9.970152 | ° 9.584177 | ~8-?8 | 40:415828 
110° Cosine, | D. 1", Sine,. | D. 1" Cotang. | D. 1". Tang. 





COSINES, TANGENTS, AND COTANGENTS BEL 





21° 


s 


WOMOsIQDOP WMH © 


10 


9 
60’ 


_ 111° Cosine. 











Sine. 


9.554329 
.554658 
.554987 
.555315 
555643 
555971 
556299 
556626 
556953 
557280 
057606 


9.557952 
558258 
558583 
.558909 
559234 
559558 
. 559883 
560207 
.560531 
560855 


9.56117 
.561501 
561824 


.562146_ 


562468 
562790 
.563112 
563433 
563755 
564075 


9.564596 
564716 
.565036 
565356 
565676 
565995 
.566314 
566632 
566951 
567269 


9.567587 
567904 
.568222 
. 568539 
.568856 
.569172 
.569488 
.569804 
570120 
.570435 

9.570751 
.571066 
.571380 
.571695 
.572009 
572323 
.5726386 
.572950 
573263 

9.573575 














Dea"; 


, 


or 
no 
oo 


= é “ neo Cae a9 19. 9 GDS Ae LPP Pee ES 
BMRB SRSNNNPVSY SSSSSSSSSS RSSRASSSKS SSSERSSESS GERKASASAES 


eo 
OW CIOTWI OT ROTRINNINIDWODODW OOCOWSCWNWWwWwwoa oo 


Or or 
ww 
Ow 





| 
_— —— 


D, 1", 








| 


Cosine. 





9.970152 
970103 
970055 
. 970006 
. 969957 
. 969909 
. 969860 
. 969811 
. 969762 
. 969714 
. 969665 


9.969616 
. 969567 
. 969518 
. 969469 
. 969420 
. 969370 
. 969321 
. 969272 
. 969223 
969173 


9.969124 
969075 
. 969025 
. 968976 
968926 
968877 
968827 
968777 
968728 
. 968678 


9.968628 
968578 
. 968528 
968479 
. 968429 
. 968379 
968829 
. 968278 
. 968228 
. 968178 


9.968128 
. 968078 
. 968027 
967977 
. 967927 
967876 
967826 
967775 
967725 
967674 
9.967624 
967578 
. 967522 
967471 
967421 
967370 
967319 
967268 
967217 
9.967166 





Sine. 











DI. 








Tang. | D. 1". 
584982 | 6-28 
“585309 ee 
585686 | 6-28 
586062 


wl 
"586439 pee 
586815 | 2-%! 

















: 6.25 
(587190 | 6-25 
587566 | 6-27 

eine vie 
vo 
"588691 aoe 
589066 693 
589440 | 6-23 
5e0814 | 6-23 
590188 | 6-23 
590562 | 6-23 
"590935 | 8-4 
6.22 
591308 | 8-22 
Fol 
592054 
"592496 | 8.20 
426 | @ 99 
592799 
OEst oe |) G20 
508171 | 6:20 
598542 | 6-18 
pos014 | 6-20 
-podees | 6-18 
‘boae56 | 8-18 
505027 | 8.18 
595398 | 9-18 
pee 6.17 
595768 
596138 pa 
596508 | 8-17 
506878 | 6-17 
seoyety | Geta 
“597616 | 6:15 
597985 | 8-13 
“pogahd | 8-15 
pog722 | 8-13 
peti pe 
599459 
590827 | 8-18 
600194 | 8-18 
600562 | 9 
6.12 
600929 | 6-18 
601296 | §-12 
‘601663 | 8 
63) 6/10 
602395 | 9: 
602761 | ©-10 
a 6.10 
608127 
603493 | 6-0? 
603858 | 6-08 
604923 | 6-08 
604588 | 6-78 
604958 | 8: 

3) 607 
605682 | 8-08 
“606046 | 6-4 

9.606410 | ° 


| Cotang. | D. 1”. 


10.415823 | 60’ 








“40.404232 








j 
Cotang. 158° 


-415445 | 59 
.415068 | 58 
.414691 | 57 
.414814 | 56 
-413988 | 55 
-413561 | 54 
-413185 | 53 
412810 | 52 
.412434 | 51 
412059 | 50 


10.411684 | 49 
-411309 | 48 
.410934 | 47 
-410560 | 46 
-410186 | 45 
-409812 | 44 
-409488 | 43 
.409065 | 42 
-408692 | 41 
.408319 | 40 


10.407946 | 39 
407574 | 38 
407201 | 37 
406829 | 36 
.406458 | 35 
.406086 | 34 
405715 | 38 
.405844 | 82 
.404973 | 31 
.404602 | 30 


7 
© 


.403862 | 28 
.403492 | 27 
.403122 | 26 
402753 | 25 
402884 | 24 
.402015 | 28 
.401646 | 22 
.401278 | 21 
-400909 | 20 


10.400541 | 19 
.400173 | 18 
.3899806 | 17 
399438 | 16 
.899071 | 15 
.898704 | 14 
. 898337 | 13 
897971 | 12 
897605 | 11 
397239 | 10 


10.396873 
896507 
396142 
89577 
895412 
895047 
894683 
894318 
893954 

10893590 


[eee ee 





a 
ao 
° 


Tang. 


a a a A PR SSR 


- 


112 


TABLE XXV.—LOGARITHMIC SINES 




















22°; Sine. 
0’| 9.573575 
1 .573888 
2 .574200 
3 574512 
4 574824 
5 .575136 
6 .DT5447 
4 .5T5758 
8 .576069 
9 .5763879 
10 .576689 
11 | 9.576999 
i .577309 
13 .577618 
14 .5T7927 
15 .5 78236 
16 578545 
17 .578853 
18 .579162 
19 | .579470 
20 9777 
21 | 9.580085 
22 . 5803892 
23 580699 
24 .581005 
25 .581312 
26 .581618 
27 . 581924 
28 582229 
29 582535 
30 .582840 
31 | 9.583145 
82 .583449 
33 583754 
34 .584058 
85 .584361 
36 .584665 
ave .584968 
38 . 585272 
39 585574 
40 .58587'7 
41 | 9.586179 
42 .586482 
43 .586783 
44 .587085 
45 .587386 
46 .587688 
47 .587989 
48 588289 
49 .588590 
50 .588890 
51 | 9.589190 
De .589489 
53 .589789 
54 .590088 
55 .590387 
56 .590686 
57 .590984 
58 .591282 
59 .591580 
60’ | 9.591878 
112° Cosine. 











Dis 


wwe 


DW SOHWSCWOHMWWMWO BDOUWVIOTNITIDNI DHODOOCWOCWNW WNWWOWOUTOOR WVIVQADDOOCCOCW 


we) Sooesooosco ScCocococoOCO0O0O COH OH He HHH Hee ee eee op et 09 0D 


S 


woMDDBDDoO 
VIIRNVRUIHHH 





re 
2 
. 











Cosine. 





9.967166 
967115 
. 967064 
. 967013 
.966961 
. 966910 
. 966859 
. 966808 
. 966756 
966705 
. 966653 


9.966602 
. 966550 


966188 
966136 


9.966085 
. 966033 
. 965981 
- 965929 
. 965876 
965824 
965772 
. 965720 
. 965668 
. 965615 


9.965563 
. 965511 
. 965458 
965406 
. 965353 
. 965801 
965248 
. 965195 
. 965148 
. 965090 


9.965037 
964984 
.964931 
964879 

- ,964826 
964773 
964720 
. 964666 
.964613 
. 964560 


9.964507 
964454 
964400 
964847 
. 964294 
. 964240 
. 964187 
9641383 
964080 

9.964026 


Sine. 








Dy 1"! 











Tang |p: 





9.606410 
606773 
607137 
607500 
607863 
608225 
. 608588 
608950 
.609312 
609674 
:610086 


9.610897 
.610759 
.611120 
.611480 
-611841 
.612201 
612561 
-612921 
-618281 
.618641 


9.614000 
. 614859 
614718 
615077 
615435 
615793 
.616151 
.616509 * 
616867 
617224 


9.617582 
.617989 
618295 
. 618652 
.619008 
619864 
.619720 
620076 
620482 
620787 

9.621142 
621497 
- 621852 
622207 
622561 
622915 
623269 
623628 
623976 
- 624330 


9.624683 
- 625036 
625388 
625741 
626093 
626445 
626797 
627149 
627501 

9.627852 

















o 


Cotang. 


ile 


So SoeoSoeSsoSeSsSsS 


ooo 


DDD WWD DOMODOOOSDD OOWODOOHDOOO OWOWOOHOOOO OOO 


NIINNINIDWID DODOCSCCOCNWNWN WO 


0 00 00 
or 


ar 





> 
09 G2 02 09 CO OT CO OT ZOTAR AIM DC SSSSSESSES 0O CO G9 CO CO OT GO CT OT 2 OF 





F 
Cotang. 157° 


10.393590 
898227 
392863 
892500 
892137 
891775 
.3891412 
.3891050 
- 390688 
.890326 
. 389964 


10.389603 
889241 
. 888880 
888520 


.888159 |- 


387799 
887439 
887079 
. 886719 
. 886359 


10.386000 
3885641 
885282 
884928 
384565 
284207 
. 883849 


883491 | 


883138 
882776 


10.882418 | 


382061 
3881705 
. 881348 


380992 


880636 
380280 
.879924 
3879568 
879213 


10.378858 
378508 
378148 
377793 
3877489 
377085 
3876731 
3876877 
.376024 
375670 


10.375817 
.374964 


“372499 
10. 372148 





Tang. 


rw) 
for) 


| QRwOR OA IMO 





67° 


a cree a RSET tern RT 


‘4 


COSINES, TANGENTS, AND COTANGENTS 118 





23°] Sine. De". 


9.591878 4.9% 


0’ 
1] .592176 
2] (592473 he 
By cnb02770 |) Fone 
4| 593067 | 4:2 
067 | 4°93 
5 | .592362 | 4:93 
6 | 593659 |, 4: 
39 |. -4°93 
7% .593955 4.93 
8 | 1594951 | 4-323 
9| 594547 | 4:% 





10] 1504842 | 4-92 





11 | 9.595137 
13 | 1595482 pov 
13 | 595727 | 4°02 
14) .596021 | 4:20 
15 | 596315 | 4°33 
'16.| .596609 | 4:20 
17 | 1596903 | 4-20 
19 | 1597490 | 4-90 
20 | 597783 | 4:30 
21 | 9.598075 
92 | 598368 ace 
23 | 1598660 | 4:8¢ 
a4 | 5oso52 | 7:8! 
25 | .5g9qda-| 4-80 
26 | 599536 | 4-8! 
a7 | .pooee7 | 4-8 
28 | .600118 ra 
29 | .600409 485 
30 | .600700 4.83 
31 | 9.600990 
32 | _601280 pee 
33 | l601570| 7:3 
84| .601860 | 4-93 
35 | 602150 | 4:& 





36 | 1602439 | 4°83 
a7 | .602728 | 7-52 
38 | 603017 | 4°32 
39 | .603305 | 4:80 
40 | 603594 | +: 


41 | 9.603882 
42 | .604170 | 4-82 
43} .604457 es 





44) ‘604745 | 4-80 
45 | .605032 ra, 
46 | .605319 | 4°» 
47 | 605606 | ine 
48 | .605892 | {+g 
49] 606179 rae 
50 | .606465 4 
51 | 9.606751 | 4 » 
B2 | .607036 | 4onn 
53] .607322 Awe 
54) 607607 1 
55 | .607892 ra 
56 | 608177 | 443s 
BY | .608461 | fing 
58 | .608745 473 
59 | .609029 | - 473 
60’| 9.609313 | .*° 





113° Cosine. | D. 1". 








Cosine. 





9.964026 
963972 





963542 
963488 
9.963434 
963379 
963325 
963271 
963217 
963163 
963108 
963054 
962999 
962945 
962890 
962836 
_ 962781 
962727 
962672 
962617 
962562 
962508 
962453 
962398 


9.962348 
. 962288 
962233 
-962178 
962123 
- 962067 
- 962012 
-961957 
. 961902 
.961846 


961791 
“961735 


ie) 


© 


Desk”, 








.961680 
.961624 
.961569 
961513 
.961458 
961402 
.961346 
. 961290 


9.961235 
.961179 
961123 
. 961067 
961011 
960955 
- 960899 
. 960843 
. 960786 

9.960730 








Sine. 





| 
Tang. D. 1". | Cotang. 156° 








— |} — 




















9.627852 | , g. | 10.872148 | 60” 
eel oe |e 
“628905 | 2-89 "371095 | 57 
"629255 | 2-88 370745 | 56 
“00608 5.85 "370894 | BD 
"620956 | 2-83 "970044 | 54 
"630306 | 2-83 "369694 | 53 
“620656 | 2-88 "369344 | 52 
"631005 | 2-82 "368095 | 51 
631 5.83 e 
631855 | 3:83 "368645 | 50 
9.631704 | 5 g4 | 10.368296 | 49 
oe ee | eee 
"632750 | 2-80 "367250 | 46 
“gga099 | 5-82 "366901 | 45 
aL 5.80 "366558 | 44 
“ea37g5 | 3-80 peer a 
fe re (er /9) 5 80 366205 43 
“634143 | 2-8 "365857 | 42 
“634490 | 3-28 "365510 | 41 
7634838 | 2-8) "365162 | 40 
9.635185 | 5 ng | 10.964815 | 39 
ee eee 
“636226 | 2-48 "363774 | 36 
"o36572 | 2:08 "363428 | 35 
‘636919 | 2-48 | “363081 | 34 
“637265 | 2: £¢ "362735 | 33 
“e37e11 | 2-f6 "362389 | 32 
“637956 | 2-5 "362044 | 31 
“g3g302 | 5.77 "361698 | 30 
5.15 
9.638647 | 5 75 | 10.361853 | 29 
ee | 68 | Bae | 
“e3g6sg | 2-05 360318 | 26 
‘640027 | 2-42 "359973 | 25 
-eu0a7i | 5-73 | “S2ee99 | St 
"640716 | 2-6 "359284 | 23 
640716 | 5 as : ¢ 
641060 | 2-4 "358040 | 22 
641404 5 72 .858596 | 21 
Ry eae "358253 | 20 
9.642091 | 5 a» | 10.257909 | 19 
642434 | Po "457566. | 18 
Bary | Os 357223 | 17 
643120 | Pe "356880 | 16 
ay 4 9 . sal or 
ee | | Bee | 

5.70 -35619¢ 
644148 | 2-4 "355852 | 13 
644490 | 2° 70 3855510 | 12 
644832 | 210 "355168 | 11 
645174 |B: 354826 | 10 

9.645516 10.354484 | 9 
645857 | 3:08 "354143 | 8 
Sn Gat | ales 
Hea |e ale 
ie) oe eee 
"648243 wes "351757 | 1 
9.648583 | > 10.351417 | 0° 








Cotang. D. 1", Tang. |66° 


rr a 


114 TABLE XXV.—LOGARITHMIC SINES 







































































{ 

244 Sine. D. 1". || Cosine. | D. 1". |; Tang. | D.1". | Cotang. 155° 
ee Pe ee SONY Bergson ot) RR 

0”| 9.609313 | 4 na | 9.960730 | 9.648583 |. @ | 10.851417 | 60” 
1| ‘609597 | 4:73 960674 | +23 648923 | 2-00 "351077 | 59 
21} .609880 473 960618 "95 -649263 | Fe. 850737 | 58 
3 | ‘e1oi6d | 4°43 960561 | “92 649602 | °° 350398 | 57 
4| veto | 4-73 960505 | *83 649942 | P-67 "350058 | 56 
5 | ‘etore9 | 4-2 960448 | °23 650281 | 3°B2 "349719 | 55 
6 | ernie | ge 960392 | -?2 “650620 | 3-65 "349380 | 54 
7 | ‘etiae, | 4-20 960835 | “92 650959 | 7-68 "349041 | 53 
8 | 611576 | 4:70 960279 | -92 651207 | eens "348703 | 52 
9| 1611858 | 440 960222 | 78 651636 | 2:62 349364 | 51 
10 | ‘612140 | 4°49 960165 | °22 651974 | 2:62 "348026 | 50 
11 | 9.612421'| 4 gg | 9.960109 os || 9-652812 | ~ 5 | 10.847688 | 49 
42 | (eiavo2 | 3:88 960052 | °?? “65850 | Fees "347350 | 48 
13 | -61g083 | 4°68 959995 | “pe 652088 | 3-63 "347012 | 47 
14} 618264 | 4-88 950088 | a2 || essaas | 2-8 "346674 | 46 
15 | 613545 | 4-68 ‘g5ggs2 | 98 "653663 | 2:6 "346887 | 45 
16| 613825 fi 959825 | +> "654000 vie "346000 | 44 
17 | 614105 | 4.8 959768 | “ee 654587 | 202 "345663 | 43 
18 | 614385 | 764 959711 | “9° 054674 | 3-62 "845326 | 42 
19 | 614665 | 4-8! 959654 | “2 655011 | 2-62 "344989 | 41 
20 | 614944 | 7°02 959596 | -2e 655348 | 2°62 "344652 | 40 
1 | 9.615228 ~ || 9.959539 - | 9.655684 0.344316 | 39 
> ss ner 959482 - "656020 | ee ‘ i icoen 38 
93 | 1615781 | 4:60 959425 | +22 -656856.| 2-60 "343644 | 37 
24) °616060 | 4-60 959368 | “22 656692 | 2-60 "343308 | 36 
25 | 1616338 | 4-63 959310 | > Be “e57ons | 2-60 "342972 | 35 
26| .61ecie | 4-63 959253 | “20 e573ed | 3-60 "342636 | 34 
27 | 616804 | 4-63 959195 | “pe 657699 | 2-28 "342301 | 33 
ps: Lereyee | 03 1959138 | -92 -e580B4 | 2-38 "341966 | 32 
29 | 617450 | 4°63 959080 | “pt “658369 | 2:28 "341631 | 31 
80)|eive7 | 4°58 959023 | “po 658704 | 2-28 "341296 | 30 
81 | 9.618004 | 4 go || 9.958065 95. || 9-659039 | 5 x» | 10.840961 | 29 
a2 | eisest | 4-6 958908 | *9> 659873 | 3-7 "340627 | 28 
33 | 1618558 | 4-6 ‘958850 | 22 "659708 | 2: "340292 | 27 
34} |618884 en 958792 eA 660042 eae "339958 | 26 
a5 | 619110 | 4-60 958731 | +2! 660376 | p24 "339624 | 25 
36] 619386 | 4-00 958677 | ° po 660710 | 2? "329290 | 24 
87 | .619662 | 4-70 ‘958619 |. 720 661043 | °P2 "338057 | 23 
38 | .619938 4.58 958561 ‘oy 661377 | Fen 338623 | 22 
39 | 620218 | 4-8 958503 | “Bt 661710 | 2°22 "338290 | 21 
40 | .620488 | 4-78 958445 |< At 662033 | >? 37957 | 20 
41 | 9.620763 | 4 xo || 9.958887 gv || 9-662876) ~~ | 10.887624 | 19 
42 | .621038 | 7°28 958829 |“). 662709 | 9: "337291 | 18 
43 | 621313 | 4p 958271 | +90 663042 eee 336958 | 17 
44 | (621587 | 4-p4 958e18 | 18! 608875 | 2-23 "336625 | 16 
45 | e2ise1 | 7B 958154 | -38 663707 | 3-73 336293 | 15 
4p | .esnia5 | Ape 958096 | +34 664089 | 2-2 "335961 | 14 
47 | 1622409 | 4-3 958038 | 34 664371 | ?°P3 "335629 | 13 
8 | 1622682 | 4° 957979 | <28 664703 | 2-23 "335297 | 12 
49 | 1622056 | 4Pe 957921 | 84 “665085 | "PB "334965 | 11 
50 | .623220 | 4p 957863 | = 665366 | P's "334634 | 10 
51 | 9.628502 | 4 x5 || 9.957804 7 || 9.665698 | 5 x5 | 10.884302 | 9 
Be | eearrd | Fee 957746 | °28 666029 | 9-3 "333971 | 8 
B38 | 624047 | 7°32 957687 | “og 666360 | P-Ps "333640 | 7 
B4] 624819 | 4°p3 957628 | -6° 666691 | 2-28 "333309 | 6 
BS | 1624591 | 4-D2 957570 | +88 667021 | 2-P0 "332979 | 5 
B6 | 1624863 | 4-P3 copra | es \oarase || Bens "339648 | 4 
By loabias | 4°33 957452 | -9 667682 | BP "339318 | 3 
58 | 625406 | 4'p3 957393 | *98 668013 | 2°35 "331987 | 2 
59 | 625677 | 4°ps 957835 | +p 668343 | 2-00 "331657 | 1 
60’! 9.625948 | 4:5" || g o57a76 | - 9.668673 | > 10°331327 | 0° 
114°Cosine. | D. 1". || Sine. | D. 1". || Cotang.|D.1". | Tang. [65° 








eee 


COSINES, TANGENTS, AND COTANGENTS 115 





25° 


0’ 


© DI Ss OTP C9 0D 


10 


115° Cosine. 











Sine. 


9.625948 
626219 
626490 
. 626760 
627030 
. 627300 
627570 
.627840 
628109 
628378 
628647 

9.628916 
.629185 


631326 


9.631593 
.631859 
6382125 


. 682392 


682658 
. 632923 
. 683189 
638454 
-633719 
633984 


9.634249 
634514 
634778 
635042 
- 635306 
635570 
635834 
. 636097 
. 636360 
636623 


9.636886 
637148 
637411 
637673 
637935 
638197 
638458 
688720 
638981 


639242 


9.639503 
.639764 
640024 
640284 
640544 
640804 
.641064 
641324 
.641583 

9.641842 


oe or on ox Sex Or 
CoO OOWW 





C2 OCLOTOUSS OF AIBN 





WD CO WOW 


wWwwwwwwwo wwwwrffl eo & 


WCOoWWwWWWWwWWwWW oo 
@O OO Go Oo OH 09 09 09 OT RRA SRSSSRZS BADDADOOCO9Oe 





AAAS A AAA. AALAAL ALAA AAS AAD AA AA PAA ALAA DAR ARERR RR AAA RRR, 








Cosine. 


9.957276 
957217 
957158 
.957099 
957040 
. 956981 
956921 
. 956862 
- 956803 
956744 
956684 


9.956625 
.956566 
956506 
956447 
956387 
956327 
956268 
956208 
956148 
- 956089 


9.956029 
955969 
. 955909 
955849 
955789 
955729 
955669 
. 955609 
955548 
. 955488 

9.955428 
955868 
955307 
955247 
. 955186 
955126 
955065 
955005 
954944 
954883 


9.954823 
. 954762 
954701 
954640 
954579 
.954518 
954457 
954396 
954385 
954274 


9.954213 
954152 
954090 
954029 
953968 
953906 
. 953845 
953783 
953722 

9.953660 








Sine, _ 





D. 1". || 


Lt cee ce A ee a eR ee a ee ee ee ee ee ee 
iS 
—) 


| 
| 


D. 


La 
aE 


| 
Tang. D. 1". | Cotang. 154° 


























9.668673 | 5 4 | 10.381827 | 60’ 
tee | oe | ate 
"669661 | 9-48 "330339 | 57 
“369991 5.50 "330009 | 56 
67 5.48 "329680 | 55 
“e70bio | 5-48 "329851 | 54 
“evog77 | 2-47 "329028 | 53 
“671306 | 2-48 "398694 | 52 
"671635 | 2-48 "328365 | 51 
671035 | x 47 828365 
671963 5 AY 828037 50 
9.672201 | x 4, | 10.827709 | 49 
612619 | 34 "327381 | 48 
meoa7 | Bae “821053 | 47 
673929 | 3-45 326071 | 44 
674257 | 2-47 "395743 | 43 
674584 | 2-49 "395416 | 42 
674911 | 2-45 325089 | 41 
EOE 5.43 a ~ 
675237 5 45 -824763 40 
9.675564 | - 49 | 10.824436 | 30 
ete | eae 
"676543 | 5:48 "393457 | 36 
“eveseg | 2-48 "393131 | 35 
"677194 es "322806 | 34 
677520 | 3-43 "392480 | 38 
vorsrrt | 8-42 | “pore | 31 
678496 | 3°45 "321504 | 30 
9.678821 | , 4o | 10.321179 | 29 
eae | 3G | Boe 
679795 | 2-40 "320205 | 26 
and 5.42 : Z 
|b | ee 
444 | 9 "3195 
444 | 5°40 
ere oa) | eS 
681416 | 2:49 "318584 | 21 
5.40 
681740 | 249 "318260 | 20 
9.682063 | 5 49 | 10.317987 | 19 
ole aaa 
"683033 | 2-38 "316967 | 16 
683356 | 3-38 "316644 | 15 
- 683356 » 538 
683679 | 2°30 "316321 | 14 
fal oe | cnee 
“eg4646 | 23% "315354 | 11 
5.37 
684968 | 2°36 "315082 | 10 
9.685200 | 5 97 | 10.314710 q 
Ol ees 
‘essoat | 232 | ‘314066 | 7 
686255 | Drs | .818745 | 6 
es6577 | 23% | ‘sisdes | 5 
686898 | 5-3 | ‘313102 | 4 
687219 | 2-3? | (312781 | 3 
687540 | 8:35 | (319460 | 2 
687861 | 5-85 | ‘312189 | 1 
a d 5.35 E : / 
9.688182 10311818 | 0 
Cotang. ! D.1’ Tang. (64° 





116 


TABLE XXV.—LOGARITHMIC SINES 








26° Sine. 
0’| 9.641842 
1 .642101 
2 . 642360 
3 . 642618 
4 .642877 
5 .648185 
6 . 643393 
7 . 643650 
8 . 643908 
9 . 644165 
10 . 644428 
11 | 9.644680 
12 . 644936 
13 .645193 
14 . 645450 
15 .645706 
16 . 645962 
ibe .646218 
18 . 646474 
19 | .646729 
20 | .646984 
21 | 9.647240 
22 .647494. 
23 . 647749 
24 . 648004. 
25 . 648258 
26 . 648512 
27 . 648766 
28 . 649020 
29 . 649274 
30 .649527 
31 | 9.649781 
32 . 6500384 
33 . 650287 
34 . 650539 
35 . 650792 
36 . 651044 
37 .651297 
38 .651549 
39 .651800 
40 .652052 
41 | 9.652304 
42 . 652555 
43 . 652806 
44 . 653057 
45 653308 
46 . 653558 
47 | .653808 
48 . 654059 
49 . 654309 
50 654558 
51 | 9.654808 
52 . 655058 
53 . 655807 
54 .655556 
55 . 655805 
56 . 656054. 
fg 656302 
58 . 656551 
59 656799 


60’| 9.657047 


116° Cosine. 











Diss 





W909 GO GO Ow O.09 CD 
SBSSHESES 


cal etetetataletetatn 
. a 
@ 


ee 
t 


atevateetedtelstetetetn 


WWNWNMNWNW WWW 


4.23 


KH woorew 
(ante clas) 


Go 


18 











Cosine. 


9.953660 
. 953599 
953537 
958475 
958413 
. 953352 
953290 
958228 
953166 
953104 
. 955042 


9.952980 
952918 
952855 
952793 
952731 
. 952669 
. 952606 
952544 
952481 
952419 


9.952356 
952294 
952231 
.952168 
952106 
952043 
-951980 
951917 
951854 
951791 

9.951728 
951665 
-951602 
-951539 
951476 
951412 
.951349 
951286 
951222 
.951159 

9.951096 
951032 
.950968 
-950905 
950841 
950778 
950714 
950650 
.950586 


950522 


9.950458 
. 950394 
9503830 
950266 
. 950202 
. 950138 
950074 
- 950010 
949945 


9.949881 


Sine. 





DF1%; 


rare 
> 
vO 


ete aes 
jon) 
Oo 


Pe a et ee ed ee Re Fe ee 
Sapeta sete ain a eG a ea AS OS ES BO «onthe eet Ly ae er oe Ge Sh Sea a ae Ge Ce ee et Cae 
S33 Sr 


ra 
fo) 
~ 





© 








| Tang. 


9.688182 
. 688502 
. 688823 
689143 
689463 
689783 
-690103 
690423 
690742 
691062 
.691381 


9.691700 
. 692019 
692338 
692656 
692975 
693298 
.693612 
693930 
694248 
694566 


9.694883 
695201 
695518 
695836 
696153 
696470 
696787 
697103 
697420 
697736 

9. 698053 
698369 
“698685 
“699001 
699316 
699632 
699947 
700263 
700578 
700893 

9.701208 
701523 
701837 
T0212 
102466 
702781 
"7103095 
703409 
“703722 
704036 

9.704350 

- 7104663 
704976 
705290 
705608 
“705916 
706228 
706541 
“706854 

9.707166 








Cotang. 





Dwi*: 


Wwowore co oD do oD co OD OD OY CD OD OD 


DBDOSSNWONOSCWN NVWWOWWOWWHWWH & 


@ CO Ww WO OD 


929% Www www 
SRSS5 SGSBSRSRSES 


VMWMWWwWNw W 








IWWW WW 
Go Otto OFT SRR 


an a gcat Peak CON Ch en hn Es ie Behe Bee ie ee a CNM ie Gali tiie SIs Fae oot ba 
eo CO CO OT? 








! 
Cotang. 153° 


10.311818 
811498 
811177 
310857 
-810537 
810217 
-309897 
809577 
809258 
808938 
3808619 


10.808300 
807981 


.307662 | 


807344 
3807025 
806707 
. 806388 
806070 
805752 
805434 


10.805117 
804799 
804482 
304164 
803847 
3808530 
808213 
802897 
802580 
802264 


10.301947 
.3801631 
.801315 
3800999 
800684 
. 800368 
800053 
299737 
299422 
299107 


10.298792 
298477 
298163 
297848 
297534 
297219 
-296905 
-296591 
.296278 
295964 

10295650. 


295337 | 


295024 
294710 
294897 
294084 
293772 
- 293459 
293146 
10.292834 


Tang. 





rs 
CO 








+ 


COSINES, TANGENTS, AND COTANGENTS 117 














27° ~# Sine. 
0’| 9.657047 

a .657295 

2 .657542 

3 .657790 

4 . 658037 

5 . 658284 

6 658531 

¥ .658778 

8 .659025 

9 .659271 

10 659517 
11 | 9.659763 
12 - 660009 
18 . 660255 
14 .660501 
ytd .660746 
16 .660991 
17 | = .661236 
| 18 .661481 
19 .661726 
20 .661979 
21 | 9.662214 
22 - 662459 
23 . 662703 
24 . 662946 - 
25 . 668190 
26 . 663433 

= 27 . 663677 
28 . 663920 
29 .664163 
30 . 664406 
81 | 9.664648 
om 32 .664891 
=) 33 - 665183 
34 . 665375 
B35 .665617 
86 | .665859 
Be or . 666100 
—688 . 666342 

_ 39 - 666583 
40 -666824 
41 | 9.667065 
42 .667305 
43 . 667546 
44 - 667786 
45 . 668027 

= 46 . 668267 
AT . 668506 
48 . 668746 
49 - 668986 
50 669225 
51 | 9.669464 
52 .669703 
53 .669942 
54 -670181 
55 .670419 
56 .670658 
57 .670896 
58 -671134 
59 .671872 
60’ | 9.671609 
117° Cosine. 














D. T". 


s 


4.07 


WOOGOSOSOOOS SESSOHSSOSOS SSSSSRSRSSeE SBBEZ 


o 
c 


WSO HWONWNWOWWWwWWWWOor Wore 


DINVINDNDWAD HWSO 





Cosine. 


9.949881 
- 949816 
949752 
949688 
9496283 
949558 
949494 
- 949429 
. 949364 
949300 
949235 


9.949170 
949105 
949040 
948975 
948910 
948845 
. 948780 
948715 
948650 
. 948584 

9.948519 
948454 
. 948388 
948323 
948257 
948192 
948126 
- 948060 
947995 
947929 


9.947863 
947797 
947781 
-947665 
.947600 
947533 
- 947467 
947401 
947335 
947269 


9.947203 
947136 
947070 
947004 

* 946937 
.946871 
- 946804 
946738 
946671 
946604 


9.946588 
946471 
946404 
- 946337 
- 946270 
946203 
946136 
946069 
. 946002 

9.945935 





| Sine. | D. 1". |! Cotang. | D. 1". 








Dit. 





ph pe ee pe eR RR et ee ep pp 
n es 
oO 











Tang. 


9.707166 


07478 
-V67790 
708102 
- 708414 
- 708726 
709037 
709349 
709660 
709971 
110282 
9.710593 
710904 
711215 
711525. 
711836 
~712146 
.712456 
712766 
718076 
718886 
. 713696 


le) 


716477 


9.716785 
717093 
717401 
717709 
.718017 
~718325 
- 718633 
718940 
.719248 
719555 


9.719862 
720169 
720476 
- 720788 
721089 
721896 
721702 
. 722009 
(22815 
. 722621 
(22927 
728282 
- 728588 
. (28844 
24149 
(24454 
. 724760 
725065 
. 725370 
9.725674 


< 








} 
D. 1". | Cotang. 152° 














5 99 | 10.292834 | 60° 
eek "999522 | 59 
Hes "292210 | 58 
oe "291898 | 57 
oi "291586 | 56 
Ba "291974 | 5B 
oa "990963 | 54 
oa "290651 | 53 
oe "290340 | 52 
eae 290029 | 51 
See "989718 | 50 
= 18. | 10-280407 | 49 
ae "289096 | 48 
bap "988785 | 47 
aes "288475 | 46 
bar 288164 | 45 
bay ‘987854 | 44 
bay ‘987544 | 43 
Bat “987994 | 42 
Day "996924 | 41 
ae "286614 | 40 
10.286304 | 39 
aa "985995 | 38 
Boe 1985686 | 37 
Be 285376 | 36 
oa 285067 | 35 
Bk "984758 | 34 
cee "984449 | 33 
Bet "984140 | 32 
ae 283832 | 31 
eee "283523 | 30 
«13 | 10-283215 | 29 
oan "932907 | 28 
oa "989599 | 27 
ae "981983 | 25 
oe "981675 | 24 
oe "981367 | 23 
See 281060 | 22 
cae 980752 | 21 
aes ean - 
Hes ‘979831 | 18 
eee “o79524 | 17 
ae “O79217 | 16 
oan "978911 | 15 
He 278604 | 14 
rs "978208 | 13 
oa “o77901 | 12 
cae 277685 | 11 
an "277379 | 10 
saiee} See ee 
5.10 ey 
Bap "o76462 | 7 
ee "276156 | 6 
oa 275851 | 5 
a "275546 | 4 
ee 975240 | 3 
oa "974935 | 2 
pee "974680 | 4 
10274326 | 0° 
Tang. |62° 


ee ge 


118 LE 
_ TABLE XXV.—LOGARITHMIC SINES 






























































28° Si n ’ 
ine. ipeaive Cosine. | D. 1". Tan | ' 
— g. | D.1". | Cotang. 151° 
0’) 9.671609 saree ila cg | 
1 | 267 3.9 9.945935 ee ~ conten 
1] 671847 | 3-95 || «245868 1.12 || 9725674 | 5 og | 10.274826 
Bl sraseneg |: BOB MILT enaeece ‘a5 “TaORN 508 | 2/402 9 
4| .672558 | 8-95 -945733 ; ROG 507 273716 | 
yee “726892 | 2-0¢ 278412 | 57 
Bile Spans.) bebe ott tear 1.18 wor 273 | 
iy 3727197 5.05 : 108 | 56 
7 | .673268 | 3-98 9455381 | 3-12 tees Ue PN "272803 | 55 
i gee. (jo met) gear 11g (||) Se 1 s.0 272499 
9 nar, oe . 945396 1.13 (27805 $ i $ © 54 
10 | censor? | 3 68 || .o4ae8 | 1.18 728109 | Bh ori ce 
673977 | 3-03 |) 945261 | 445 “sis | 5:05 | “brie | Bi 
11 | 9.674213 | ee Ck Seemed Dee aa ie MBBS | 1 
12 | e7444g | 3-92 9.945193 9% 5.07 -271284 | 50 
He envie, |S mn oH) Cee 1a ee a te 10.270980 
14 674919 3.92 Sayin pe esi ite 270677 i 
5 | (675155 | 3-93 || | 990 | 1-1 teeeee HhS is 270874 
el ie "730233 | 0-07 270071 | 46 
‘ "675624 | 3-90 944854 | 1-18 "3 5.08 269767 | 45 
18 | ‘e7sss9 | 3-92 944796 | 1-18 130535 | 5°95 269465 ‘ 
18 | 675850 | 3 |) -24t18 | 113 vraosas | 5-03 | “oeoie | 43 
hickuees Cie allie 44050 1.18 |) ae oie “beseso | 42 
a1 | o.erese2 | 2 II 9. B82 | 143 || -731746 | 2-3 Pome 
Sel Apeenso | (8180 ‘944446 | 1-18 (32048 | 5 ox | 10.267952 
| ‘ez7a98 | 3-90 -944309 | 373; ‘wapees | DOS 267347 | 37 
il Beran | 8 eee ll Pansat 1:18 || iSaeer | 5. 2670 
sag Pe lal 441% 8 i] i7gge57 | 5-98 .267045 | 36 
Greene: | PBB el aatind pis 733 5.02 266743 | 35 
28 | le7gi97 | 3-88 944104 | 1-13 Jae SD 266442 | 34 
29 “678 3.8 ‘944036 | 1-18 33860 | 5" . ae 
30. “678063 | 3 BS |) c9a067 | 3:15 | 734162 | Pop "Be5es8 | 32 
3-8 ||) x9aseno | 338 ‘teqae3 | 5-02 | “besser | 31 
31 | 9.678805 |... 1B ||. raves | Oo 65587.) 31 
32 | ‘679198 | 3-88 9.943830 | |. 9.43 5.08 265236 | 30 
33. 679360 3.387 . 943761 1.15 ae: 5066 5.0 10. 264934 
Si “onopme | 387 |) sito as || eee | sc) | | eae 28 
5 | 67982 3.87 . 943624 Oi “nekae ; : Oi tag 
stone | ES |) BBs | 3p ‘oan | 25 | “aoBr | 
7 | ‘6g0288 | 3-87 943486 | 1-1 9 5 02 "263731 | 25 
Soccaeuesg | (B88 oll amuenas 1345 Cll eee st 263430 
Pee haeey BB. Beaepes tae || -736870 | Bop | 268 a 
40 ; 680982 3.87 .94827' 1.15 ; exete : { Kd f oes ae 
385 :943210 | 1.15 BCATT | 2625 _ 
41 | 9.681213 | 5 oo Lae Reamer Lea nha 29 | al 
42 | (681443 | 3-83 9.943141 9.3 5.00 .262229 | 20 
43 681674 3.85 . 942072 1.15 de 8071 5.0 10.261929 
pe "681905 | 3-85 a : - hit ett 6 2 261629 Ae 
i "682195 | 3-83 . 942034 : “93 : 5.00 261329 | 17 
46 | 1689365 | 3-83 oaaped | 1-17 “feet, | 5.0 261029 
ar | esas | (BIRR S| Ceaeene Tis || 739871 | fgg | 260720 3 
Tel Repsebe | BBO Heaeate iis || 73957 | $90. | -260480 id 
49 ee APS: te IMs Milt waaeras Bp a 
Sot, teeaaee 3 Be 087 — ie 498 250831 | 12 
Bi | eceasead 3.83 942517 | ys “naorey | 4:98 259582 | 11 
bs ores | Bao] Seaeaees 9 4.98 .250233 | 10 
53 | egag72 | 3-82 ‘940878 | 1-12 741066 | 4 10.258934 
Bi) sient | 38) 942308 | 1-15 1086 | 4s | | aes : 
erish | (Bite Ol! Rete eels a7 4.97 "258336 | 7 
56 | a4658 | 3-80 “oaoie9 | 1-17 (41962 | 4’ 258038 
me \ceasiegy.| 8182 || bea At || e42e6i | fsbo | eaerrno 6 
Ne cacetis | 8.80 ‘942029 | 1.1% WAR559 |g" a “OBrd4t ° 
Bel Gingegin | (8:80.51) | eemaees 11 || rapes | 4:88 | lapy ss 
| gee | 233 | SMe | at | | et 2s | 2 
Pa a! Somme as ; 1 : . ‘ 
ee cei ial a 1819 9.743752 | 4:97 Paso 1 
sine, D. 1’. Sine pea ee Seay 256248 0’ 
. 1". || Cotang. | D. 1" Tang. 61° 











COSINES, T 
, TANGENTS, AND COTANGENTS 119 






























































29°] Sin «| 
e. D. 1”. || Cosine. | D. 1’ : 
¥ foe Ee Tang. Daz | 
. 1", | Cotang. 150° 
0’| 9.685571 Leh ap [ees 
H 685799 aye iH Gaitean | Gar a 02 
) i | ; 7 ; ae OIVDn 
4| eseas2 | 3-80. ‘941609 | 1-27 TH | aos ‘sist 38 
5| 686709 | 3-4 || "944539 | 1-1% Lie ri 588s | 
6| .686936't 3-4 “oaj4e9 | 1-17 Mp 195 Sonnet 38 
o | ‘es7i63 | 3-48 “o1i308 | 1-18 745240 | 4-99 B60 | BB 
g| eszasg | 3-4¢ 941328 | 144 aes | ule x 
9| ‘es76ig | 3-23 | guess | 4th HES | Las eu : 
10 | ‘es7s43 | 3-0 o4ti87 | 4°79 T6429 £95 ass | 
se fe | 1a T6132 | 495 253868 | 52 
11 9. 688069 28 | a | 2 set 
12| _ease05 | 23-02 tego 49 ear 
Be becca: |) 82?" || iceanpos 148 |" taraio 352081 
14| less7a7| 3-2 940905 | 3°44 Tt616 18 3 | § 
15 |  esgg72 | 3-6 ‘940834 | 1-18 AarO13 4:95 oat | 
16 | .6ggi98 | 3-20 940763 | 1°5 Moe | 43 pe i 
i7| 689423 | 3-4 940698 | 346 ewe | 49 ae 
18 | 6gg648 | 3-4 e462 | 748 en | 4% ue : 
19| 689873; 3-4 | 940851 | 1°48 “Hoe ‘9 ae 3 
a9 | 690098 | 3-22 940480 | 7-78 fae | sms | 8 
oe SOR) | 1 “749097 | 4°93 "950903 | 42 
21 9. 690323 3 v5 || 9.940338 i bce i 2 “ 
23 pees 37 ‘osoge7 | 1-18 pees @ | 08 a8 
04 690996 | 3-23 "940196 | 1-18 tah 19 2 va) 88 
95 | 691220 | 3-23 ‘940195 | 1-18 ee 41.93 inks a 
96 | 691444 | 3-23 "940054 | 1-18 ae 192 Si0ies * 
o7 | ‘691668 | 3-73 939082 | 1-43 “Dole n us : 
98 | ‘egia92 | 3-3 oso11 | 3748 “lier mn 3S 4 
09 | 692115 | 3-2 -939840 | "3p Buns | 4. 38 | B 
39 | 1692339 |. 3-43 930768 | "ip peat im 3 me 
| oo) || *onadey | tees foe | i | Bs 
31 | 9.692562 i ene | i a | 
30 | 692785 | 3-/2 a te 19 2m 2 
33 | 693008 | 3-44 089554 | 7148 poet | 4 ab 
34 693231 | 3-72 “939482 | 1-20 “ean rr ae 
Be eieaicg |: 3590" ||), conseay 130 || FEBS | 4 Se 
3¢| 693676 | 3-02 939839 | 1:28 “TpAlis be : a | 
37 | 693898 | 3-20 939267 | 1" “T5409 0 ae = 
33 | ‘694120 | 3-7 "939195 | 1-20 7954400 || “450 2 pol 2 
39 | 694342 | 3-20 "939123 | 1-20 T5403 | 9p Sts; E 
Pe cohasas | 8:20"|| < beeoes ie |) fear | 4m ee 
41 3°79 || +988980 1.20 T5201 | 490 Buin | 3h 
ia 12 T5591 | 499 "244709 | 21 
42| 695007 | 3-88 ae ie 490 2429 
8) ists i 988858 | 156172 4.90 Od 122 19 
| $885 | 2a | ses | 122 "756465 4.88 A oe 18 
i ‘ssi 3s 038601 Ee (56759 4.90 aa 17 
47 | (696113 | 3-88 938547 | 4:5) eye 1s pe i 
tea ts 988647 Le I5TBAS 4.88 ; Se 15 
49 | 696554 | 3-87 938402 | "55 eT 18s pe : 
F ‘ 3 938402 i T5798 4.88 sa 362 | 13 
; ofp 1 ||) soneess | 1-20 TERE | es aur 
51 | 9.696995 ee sles Se | 
52 | 697215 | 3-9¢ “eeetis ole 47 am 9 
Be Ong | 3.0 938185 | 99 158810 10.24 
54| 697654 | 3-85 ‘93gq40 | -1-22 ‘cena 183 S088 
55 | 697874 | 3-82 ‘o3v9e7 | 1-22 150687 181 aie | 4 
56 | 698094 | 3-87 || “937995 } 1-20 Heed ra Sian 
By | 698313 | 3:95 vearsee | 1-82 fae idee a 
| ee] Fe oorene | 1% “Toei 4.88 Sete es 5 
% te 3 937749 Le ; 76056 4 4.87 4 td 728 4 
ee anion | 18 eee eae "930436 | 3 
ir 9 i604 | ibe || OILS re Speraeln4 
: ae ‘s : Fs ow 
119° Cosine. | D. 1’. Sine DEX | ——— — : 
A bee Cota avy es 60° 
ng.!' D.1". | Tang. !60° 





120 


TABLE XXV.—LOGARITHMIC SINES 





30° 





20 


— 


120° Cosine. |! D. 1". 


et fee fe fel ped fee fed fret fh fod 
SOIAARAOE Somme | 











Sine. 


— 


9.698970 
.699189 
.699407 
699626 
.699844 
- 700062 
. 700280 
.700498 
100716 
- 700933 
701151 


701368 
. 701585 
.701802 
.702019 
. 702236 
102452 
702669 
. 702885 
703101 
703817 
9.703533 

703749 

. (03964 

704179 

. 104395 

. 704610 

. 704825 

. 705040 

~ 105254 

. 705469 
9.705683 
705898 
706112 
706326 
.706539 
706753 
. 106967 
707180 
707393 
107606 


ie) 


9.707819 


- 708032 
708245 
. 108458 
708670 
. F08882 
. (09094 
. 709306 
. 709518 
709730 


9.709941 
710153 
. 710364 
10575 
110786 
(10997 
711208 
-711419 
711629 
9.711839 





D. 2. 


D2 D> DS? Ss S>? S2 GS? S> 


oo ce G9 G9 CO OD C9 C9 CD 
for) 
© GO G9 Co Go Oo OL Oo OT 


3.63 





























J 
Cotang. 149° 


SF |S | | | | | 








Cosine. | D. 1’. 
9.987531 
987458 es 
937385 | 1-22 
937312 | 1-22 
937238 | 1-23 
937165 | 2:28 
g37o92 | 1-28 
‘o37019 | 1-32 
"936946 | 2: 
36946 | 1°93 
oe ae 
9.936725 . 
936652 | 1-28 
936578 | 4°33 
936505 | 1-22 
936431 | 1-33 
936357 | 1:33 
7036284 | 1-28 
936210 | 4-33 
936136 | 3°53 
936062 | 1-53 
9.935988 
-o350i4 | 1:38 
“935840 | 2-2 

Rey i - 93 
a 935 ( 66 1 93 
“935692 | 1-33 
935618 | }*33 
"935543 | !- 

. 1.23 
935469 | 4-53 
“935805 | 1-33 
7935820 | 1-52 

9.935246 < 
OB51T Il | Lies 
935097 | J "5s 
7935022 | 1-23 
"934948 | 1: 

48 | 7195 
“o34798 | 2: 
oe 1.25 
9384723 1 93 
"934649 | 3-58 
984574 1.25 

9.984499 
o3ddea | 1-3 
934349 | 1-22 
o34a74 | 1-22 
934199 | 13> 
7934193 | 4-80 
“934048 | 1-8? 
933973 | } "5. 
933898 |} "Sy 
933622 | 1-3! 

9.983747 | 4. 
933671 ae 
933596 | }:5> 
“933520 | 4-82 
"933445 | 1-3 
933369 | 1-30 
“938293 | 1-30 
933217 | 1-30 

9.933066 | 1: 
Sine. | D. 1°. 




















9.761439 
761731 
762023 
- 162314 
- 162606 
. 762897 
763188 
768479 
163770 
7164061 
164852 

9.764643 

764933 

165224 

765514 

. 765805 

- (66095 

766285 

166675 

166965 

(67255 

767545 

(67884 

768124 

168414 

(68703 

- 768992 

.769281 

(69571 

. 769860 

770148 


9.770437 
T0726 
.T71015 
771803 
171592 
771880 
772168 
172457 
T2745 
7730383 

9.773821 
773608 
.773896 
174184 
4A 
TAT 
175046 
775333 
775621 
«775908 


9.776195 
776482 


ve) 


Cotang. 





oie aie olle ice ee) 
Ort Or OF OT OT OL OT 2 OUST SF 


G0 00 G2 CO CO 


i RRR RRR SLE A AA SA PALA A AAA AAA PALA AAP ARE POD 


DW-I WOWDDWDDWODMDMD WOWWOWOWMMMO®D 
Cow SSSESSESSESS ri) SEES RGR CRORE 28 28 C8 Sb ob Oo Ot oo RS 


4.80 


78 


AJA AJII 
RBZISIS 


4 
KS 





10.238561 | 60’ 
-238269 | 59 
237977 | 58 
-237686 | 57 
.237394 | 56 
.237103 | 55 
-236812 | 54 
.236521 | 53 
2386230 | 52 
-235939 | 51 
235648 | 50 


10.235357 | 49 
-235067 
234776 
.234486 | 46 
.2384195 | 45 
233905 * 
-238615 | 43 
-233825 | 42 
.2330385 | 41 
282745 | 40 


10.232455 | 39 
(232166 | 38 
(231876 | 37 
(231586 | 36 
(231297 | 35 
"231008 | 34 
'230719 33 
"230429 | 32 
(230140 | 31 
.229852 | 30 


10.229563 | 29 
1229274 | 2 
{228985 27 
.228697 26 
"228408 25 
1228120 24 
1227882 | 23 
1227543 | 22 
227255 
226967 | 20 

10.226679 | 19 
226392 | 18 
(226104 | 17 
1225816 | 16 
"225529 | 15 
1225241 | 14 
1924954 | 13 
“224667 | 12 
224379 | 11 
(224092 | 10 


10.223805 
223518 
223232 
222945 
222658 
222872 
222085 
221799 
221512 

10.221226 


Tang. 


> 
ie 2} 


i 


rw) 
pe 





oa 
> | Qt 0 CO He OTR 2 GO 





as 


31° Sine. 


0’ | 9.711839 


1/ .712050 
2| .712260 
3| .712469 
4| .712679 
5 | .712889 
6 | .713098 
7 |  .713308 
8 | .713517 
9 | .713726 

10 | .713935 
11 | 9.714144 
12 | .714352 
13| .714561 
14| 714769 
715 | .714978 
16 | .715186 
7 | .715394 
18 | .715602 
19 | .715809 
20 | 716017 
21 | 9.716224 
22 | 716432 
23. | .716639 
24 | °716846 
25 | .'71'7053 
26 | .717259 
a7 | .71'7466 
28| .717673 
29 | .717879 
80 | 718085 
-- 81 | 9.718291 
32 | .718497 
83 | .718703 
34.| .718909 
85 | 719114 
36 | .719320 
87 | .'719525 
38 | .719730 
39 | .719935 
40 | .720140 
41 | 9.720845 


42 | 720549 
43 | .720754 


44 | .720958 
45 | .721162 
46 | .721866 


47 | .721570 
48 | 1721774 





49 | .721978 
50) .722181 
51 | 9.722885 
52 | .'722588 
53 | .72279% 
54 | £22994 
55.) .723197 
56 | .723400 
57 | .723603 
58 | 728805 
59 |  .724007 |- 
60’ | 9.724210 





121° Cosine. 











es 


Cs 
Cw 














Oo BESS EDEAS Co Go C9 OF OV G9 OL OU OT 2 SRaSSSeERER Denese SSESS 











Cosine. 


9.933066 
982990 
932914 
932838 
982762 
932685 
. 982609 
982533 
982457 
932380 
982804 


9932228 
(932151 
932075 
-931998 
931921 
(931845 
(931768 
-931691 
“931614 
“981537 


9.931460 
931883 
. 931806 
931229 
.9381152 
.981075 
- 930998 
.980921 
. 930843 
930766 


9.930688 
9380611 
930533 
.930456 
9380378 
9803800 
- 980223 
980145 
. 930067 
929989 


9.929911 
929833 
929755 
929677 
929599 
929521 
929442 
929364 
929286 
. 929207 

9.929129 
929050 
928972 
928893 
928815 
. 928736 
- 928657 
928578 
928499 

9.928420 


Sine. 





D. 1". 


fh ek ee eR ee ee eR ee pe et pe et RR PP pp pt Pt et 
ri) 
[o-) 


ee 
sy) 
CS) 

















Tang. 





9.77877 
. 779060 
779346 
1796382 
779918 
780203 
. 780489 
180775 
. 781060 
. 781346 
781631 


.(81916 
782201 
~ (82486 
182771 
- (83056 
. 783341 
. 783626 
. (83910 


ile) 


“784195 | 


(184479 
9.784764 
~ 785048 
(85332 
. 785616 
785900 
. 786184 
. 786468 
(86752 
- 787036 
187319 


. 787603 
. 87886 
. 788170 
. (88453 
. (88736 
789019 
7189302 
. 789585 
. 789868 
790151 


9.790434 
190716 
790999 
791281 
791563 
791846 
.792128 
792410 
192692 
192974 

9.793256 
. 798538 
793819 
794101 
. 794383 
. 794664 
794946 
195227 
795508 

9.795789 


ive) 


Cotang. 











Dae 


a3 JAF -I-J-Y+~I 
VIN IVIIA 


AAA AA AA AAAS ADDR RRA AR AA AAD PARAS AAA AA SARS RRA AR RAR RR RRR 
=~ 
[we] 


4.68 


Dai 





COSINES, TANGENTS, AND COTANGENTS 


Cotang. 


10221226 
220940 
220654 
. 220368 
220082 
219797 
219511 
219225 
218940 
218654 
218369 


10.218084 
217799 
217514 
217229 
216944 
-216659 
216374 
216090 
215805 
215521 


10.215236 
214952 
214668 
214384 
.214100 
218816 
218532 
218248 
212964 
212681 


10.212397 
.212114 
211830 
211547 
211264 
-210981 
210698 
-210415 
-210132 
209849 


10. 209566 
209284 
209001 
208719 
208437 
-208154. 
.207872 
.207590 
207308 
207026 


10.206744 
. 206462 
206181 
-205899 
205617 
. 205336 
205054 
204773 
204492 

10.204211 


Tang. 





121 


148° 
| 





Un inEEnSnERnesiemeneteneneia nae 


ee 


34° 














124° Cosine. 


Sine. 


9.747562 
TATT49 
747936 
748123 
. 748310 
748497 
748683 
748870 
749056 
749243 
749429 
149615 
749801 
. 149987 
(90172 
. 150358 
. 750543 
750729 
750914 
751099 
751284 
9.751469 


© 


952044 
“853128 
9.753312 
53495 


9.755143 
759326 
. 755508 
. 755690 
(50872 
(56054 
. 756286 
756418 
756600 
56782 
. 756968 
157144 
757826 
57507 
157688 
757869 
. 758050 
- 758250 
758411 
§. 758591 


io) 


TABLE XXV.—LOGARITHMIC SINES 














Do 1". 


co 
= 


RIRRVVIRR 


Seoeooooes Oeeeeoeoeooso ooooeoocooso oooocoo 


SPMWOSWWNVWMWNMW WWWWWWCCWWO O69 OTOTOT O23 2-2 OF 


4 




















Cosine. 


9.918574 
918489 
918404 
918318 
918233 
918147 
. 918062 
917976 
917891 
917805 
917719 


9.917684 
917548 
917462 
917376 
917290 
917204 
917118 
- 917082 
916946 
.916859 


9.916773 
916687 
916600 
916514 
916427 
916341 
916254 
916167 
.916081 
915994 


9.915907 
915820 
915783 
915646 
915559 
915472 
915885 
. 915297 
. 915210 
915123 


9.915035 
914948 
914860 
91477 
914685 
914598 
914510 
914422 
914334 
914246 


9.914158 
.914070 
913982 
. 913894 
. 913806 
.918718 
. 918630 
913541 
. 913453 

9.913365 


Sine. 








D. 1". 


e 


et ek ee ep pe Pe pe Pe ee ee ee Pe ee ee pe Pe ep ep pe pp pp 











| 
Tang. | D.1". | Cotang. 145° 


9, 828987 
829260 
8295382 
829805 
830077 
830849 
.830621 
. 830893 
831165 
831437 
881709 

9.831981 
832253 





832525 
832796 
. 833068 
.§33339 
833611 
833882 
884154 
834425 


9.834696 
834967 
835238 
835509 
835780 
. 836051 
836822 
836593 
. 836864 
837134 


9.837405 
837675 
837946 
838216 
.833487 
838757 
839027 
839297 
839568 
839838 


9.840108 
840378 
.840648 
.640917 
.841187 
841457 
.841727 
.841996 
842266 
842535 

9.842805 
843074 
. 843343 
. 843612 
843882 
.844151 
.844420 
. 844689 
.844958 

9.845227 











Cotang. 








4.55 
4.53 
4.55 


ARADO RA ORO oOOoOp oo Cooma oF 
DODDDODDD SCOHODOSOHOS SOWSOSOWOWS DY 


Dein 





10.171013 


"168291 


10,168019 
167747 
167475 
167204 
. 166932 
. 166661 
166389 
.166118 
. 165846 
165575 


10. 165304 
165033 | 
164762 
1164491 
164220 | 
163949 
163678 
163407 
1163136 
162866 


10.162595 
162325 
162054 
161784 
-161513 
161243 
160973 
160703 
160432 
160162 


10.159892 
159622 
1159352 
1159083 
158813 
1158543 
158273 
158004 
157734 
157465 

10.157195 

156926 
9. 156657 
156388 
156118 
155849 
155580 
-155311 
155042 
10.154773 











Tang een 


m2 COM O1Co 3 HO 


0’ 







































































fe) * 
33°) Sine. Deli"; || Cosi 
ne, D as 
nbs a i ng « . Tang. D. 1’. Cot | 
0’ | 9.736109 MeL. ene 1460 
1] .736303 3.93 || 9-928591 
2| ‘736498 | 3-29 “993509 | 1-37 9.812517 sa 
3| “ge692 | 3-23 ‘993407 | 1-37 “giov94 | 4-62 10.187483 | 60/ 
4} 736886 | 3-28 "993345 | 1-37 "g13070 | 4:60 "187206 | 59 
5| 737080 | 3-23 "ga30g3 | 1-37 "gigsa7 | 4-62 .186930 | 58 
6 |  73yo74 |, 3.238 ‘g23is1 | 2-37 "913603 | 4-60 186653 | 57 
v| lvavae7 | 3.22 “gagogg | 1-38 "813899 | 4-60 .186377 | 56 
g| iz3vec1 | 3-23 “923016 | 1-37 "g14176 | 4-82 .186101 | 55 
9 | 2737855 3.23 "922933 | 1-38 1914452 | 4:60 . 185824 | 54 
10| .738048 | 3-22 "goo8s1 | 1-87 "g14728 | 4-60 -185548 | 53 
11 | 9.738241 322 “goovgg | 1-38 815004 4.60 185272 | 52 
Sees, | mia. |) eee ng, | ee pet eg a 
13 | 738627 | 3-22 “g20603 | 1-38 9.815555 |, 184720 | 50 
Meier ay | S28. || Seen eed | roegeest | 400 10.184445 | 49 
15 | (739013 | 3-22 “goa43g | 1-37 || “gigio7 | 4-60 “184169 | 48 
16 | 739206 | 3-22 "920855 | 1-88 "816382 | 4:58 .183893 | 47 
17| 739398 | 3-20 "ga00%9 | 1.38 "g16658 | 4-60 .183618 | 46 
18 | 739590 | 3-20 "g20189 | 1-38 "816933; 4-98 "183342 | 45 
19 | .739783 | 3-22 "g2210¢ | 1-88 "gi7209 | 4-60 -183067 | 44 
20} = .739975 3.20 “gao993 | 1.38 "17484 | 4-98 182791 | 43 
o1 | 9.740 3.20 “921940 | 1-88 "gi775g | 4-58 .182516 | 42 
99 | (7 167 9 1.38 "18035 | 4-60 .182241 | 41 
“40359 | 3-20 . 921857 4.58 181965 
93} ‘740550 | 3-18 “gai774 | 1-388 9.818310 ; 40 
24 | 1740742 | 3-20 “gaieg1 | 1-88 "g1g585 | 4:98 10.181690 | 39 
25 | 740934 | 3-20 “g2ieo7 | 1-40 ‘giss60 | 4:58 .181415 | 38 
26 |. 741125 318 || -92l524 1.38 "819135 | 4-58 "181140 | 37 
er | iraizie + 3-18 | ‘g2i4gi | 1-38 "gi9410 | 4-98 -180865 | 36 
98 | (741508 | 3-20 “gaign7 | 1-40 “g19684 | 4-22 .180590 | 35 
99 | (41699 | 3-18 “gato74 | 1.38 "g19959 | 4-58 -180316 | 34 
30 | .741ss9 | 3-27 “921490 | 1-49 "gaqa34 | 4-58 180041 | 33 
31 | 9.74 3:18 “oa1107 | 21-88 "gog508 | 4:5¢ .179766 | 32 
901 7 2080 9 1,40 "g20783 | 4.58 179492 | 81 
- (42271 3.18 . 921023 4.57 179217 
33 | 1742462 | 3-18 'g20939 | 1-40 9.821057 |, . 30 
34| 742659 | 3-17 ‘g20856 | 1-88 "921332 | 4.58 10.178943 | 29 
Dap | 742942 | 3-12 ‘g20772 | 2-40 "821606 | 4-50 178668 | 28 
86} .743033 3.18 “920688 | 1-40 "gaigso | 4:5¢ “178394 | 27 
37 | 743223 | 3.1% sr Seg Cr Fel Reg 4.57 :178120 | 26 
“oy CR BP ait tag. || meesendes S's 177846 | 25 
39 | 743602 | 3-15 "900496 | 1-40 “goovo3 | 4-97 ATT | 24 
40 | 743792 | 3-12 “g203n9 | 1.40 "googvy ) 4.570 177297 | 23 
4i| 9.% 3.17 “920268 | 1-40 "gago5y | 4-97 177023 | 22 
ve) . 743982 9.9 1.40 “ga3504 | 4-59 .176749 | 21 
“wadi71 | 3-15 920184 4.57 "176476 | 2 
43 | 1744361 | 3-17 ‘g20099 | 1-42 9.823798 | ce 20 
“n44550 | 3-15 "20015 | 1-40 ‘ga4072 | 4:5% 10.176202 | 19 
45) «744739 3.15 ‘gi9931 | 2-40 “go4345 | 4-85 .175928 | 18 
46 | 2744928 | 3.15 et Wa || ee 4.57 175655 | 17 
4% | 745117 3-15 “919762 | 1-49 "ga4g93 | 4-57 .175381 | 16 
48 | 745306 | 3-19 ‘919677 | 1-42 825166 | 4-35 -175107 | 15 
949) (745494 | 3-13 igen yh te i Sree 4.55 174834 | 14 
BO | 745688 3.15 “g19508 | 1-42 "ga5713 | 4:57 174561 | 13 
51 | 9.7% 8.13 “9i94a4 | 1-40 “ganggg | 4:59 174287 | 12 
5 745871 4.42 826259 4.55 .174014 | 11 
52) .746060 3 15 || 9-919339 45D "1739741 
53 | 746248 | 3-13 “g19054 | 1-42 9 826532 : 10 
54] .746436 3.13 “oi91e9 | 1-42 "26805 | 4:55 10.173468 | 9 
55] l746e04| 3-18 "919085 | 1:40 “garovg | 4:59 173195 | 8 
56] 746812 | 3-18 “919000 | 1-42 "07354 | 4-99 172922 | 7 
57 | 746999 | 3-12 918915 | 21-42 "govgo4 | 4-55 172649 | 6 
58 |  747ig7 | 3-13 919830 | 1-42 "govag7 | 4:99 (172376 | 5 
59 |. 747374 3.12 918745 | 1-42 "gogo | 4:59 172108 | 4 
60'| 9.747562 | 3-18 918659 | 1-43 "goga4g | 4.53 -171830 | 3 
—- 9.918574 | 1-42 "gog715 | 4:59 171558 | 2 
193° Cosine. | D. 1". pee fh Reeese7 4.53 "171285 | 1 
_{D.1". || Sine. | D. i", Il Cotang. 10171013 | 0’ 
. 1°, || Cotang. | D. 1’ ———_— |— 
i Tang. 56° 











124 TABLE XXV.—LOGARITHMIC SINES 


SSS ee tt a EME ee a ae 



























































ga sine. | D.1". || Cosi 
. . . osine, by ” 
Ls e. | D. 1". Tang. D. 1". | Cotang. 147° 
0’| 9.724210 9.9284: we a oe 
igo ie open 928420 9.7957 
dle ee [abe || Mga oe fein |v 4368. (Vapegnean tm 
3| .7ede16 | 3-3! perce (eee {I aie ge 468 “03849 | 58 
4 7 ‘ us ; . 96632 : \ 
5] fomg | 23% |) “opsoes 1-33 || “roeois | 43 | “Sooner | 86 
6 | ~ .725420 | 5° “govo4g | 1-32 WITADE |g” :202806 | 5 
meee es “927867 | 3-36 mopman |. 4.68 202526 | 54 
g | “795803 | 3-35 9077 1:33 T9TTS 202245 
Bl Tetceper-| 385" || apse (se "7gg036 | 4:68 |: oa 
10 es 3.35 92708 he "798316 | 4:82 aoe ee 
726225 | 3°95 || -927629 | j'33 || «798596 4.67 eee: bo 
11 | 9.726826 | 5 55 || 9.927549 23) g vaserz | 1° eer ee 
FS ry 92 - ri L: . Tai 
ra | fammes | 382 | reed | 388 | Scromar | 07 | cae | 4p 
1 Ter0% ee spetory 1 dee Lt ee omes an "200563 | 47 
5 |° .7e7ee8 | 3° ‘3 || 799717 | 4g; 
8) eae] ie || Beet) 5m | cowet| ie | ane | 
17 ey 2 ; . ee 2 lod " ne UV iY 
A | care | cas || Pet | a. RE | 10048 43 
19 "2 lod OK . 800836 
BP \aatereee |) 8688 926011 LOS) | aeeotiig pa Srtire po 
333 || -926831) 4°5 Spige, (i SO% meee ee 
a1 | 9.728427 Re A aes 4:65, | SesOeet ge’ 
35) 7asé26 | 3-82 |] “‘oager1 | 1-33 9.801675 10.198825 
2 728825 ey Peet des Wee 4°67 | “1198045 38 
4| 799024 | 3-82 || ‘ogesar | 1-88 7 2234 |g 197766 
95 729903 | 3-82 || ‘og64s1 | 1-38 ‘soasia | 4:05 | “rorasr | 36 
| : 926431 | 4° mo | 4-65 197487 | 36 
26 (729402 | 3-82 pret 188 ieepeers 
2 1926351 | 7° ae 1) 48a -197208 | 35 
97 | (729621 | 3-22 erp he 803072 196928 
98 | 72 ged Bees » "903351 | 4:65 |: pes 
oe eons | 8-30 || “geeiio 133 ‘soseso | 4-63 | “to6az0 | 33 
BLP tepete {t BiBeY lMegeeyey 1e1685. I auanaier 4.65 | “Toeoo1 | 34 
31 | 9.730415 9.925919 | 4 4.05. | Se eae 
32 | 730613 3.30 he 1.35 9.804466 10.1955 
| aeee0811 |) oy SNaeree 1 aneay Weneene es ee “195955 eM 
34 | 1731009 | 2: etl a a Oe aries Lops "194977 | 27. 
35 | 273 3.28 e5707 | 1°35 || 805802 | 4 Ge nea tae 
3B | “yaiaog | 3-80 rg tgp || -805580 | 468 “430 | Be 
37 | .781602 | 3-30 eS Wes! ae 4.65 | “agai | 4 
38 731799 | 3-28 apy 1198 806137 | 4 68 eee = 
39 | 731996 | 3-28 Beans Wee Pete gays pie “193585 | 38 
40 | 732193 | 8-28 25303 | 3 "39 “g0g693 | 4:63 -193585 | 22 
Srey |) weoebene Face MeBNOTL 1a eo ee 
EY aeloweay esl Sere hay ll g sores | Se Satta 
#2 | aores | 3-28 || “gpuoro eS | ousiios are ie toeara (ae 
44 | .732060 aes ebm Mes rae ota 4.63 T3105 | IT 
5 | vars? |) 2. “99 1.35 || -808083 | 4’ "19191" 
ue | oe ‘ 3 27 - 924816 : “8083 4.63 -191917 | 16 
4g | 733765 | 8-27 pega | 28% Weenargs tee “T0108 | 13 
49 733 ae a : } .809193 5 
paps esa |r err dal beers vies meponen tetas ae 
3 || 924409 | yge || 809748 | 4 gp ae 
51 | 9.734353 9.924328 | 4 : gee ae ie 
52 734549 3.27 oe ih ae 9.810025 10.189975 
BB | Toad ares ee iieaeet || leerpaus race 180008 | 8 
"734939 | 3. ee aiies || eer ee ea "189420 
55 | 735135 | 2-27 ae | 87 810857 | 4°63 7189183 | 6 
Bol wieccees | aces eeeenig sl) tar eras, 4.62 | “T38866 1 5 
Pe deems {SOD I geeeeay 1287 aera 4.60 |  “isg500 | 4 
retinas 393 923837 | “siiegy | 4-62 -188590 | 4 
| cfr | 88 | ere] 3 | cee] £8 | ae 3 
60’| 9.736109 | 3-25 923673 | 1°34 "s19e41 | 4-82 Speers 
ae a a 9.923591 | ~" 9 cisi7 | 480) | yoaerass hale 
122° Cosine. | D, 1’. || Sine. | ee i 
; ine. | D.1*. |! Cotang. | D. 1” Tang. 57° 











a 


ee Se 


COSINES, TANGENTS, AND COTANGENTS 125 


















































: ‘ | 
35° Sine. Div. Cosine. | D. 1". Tang. Disks Cotang. 144° 
0’| 9.758591 9.913365 9.845227 10.154773 | 60° 
1| 7872 | 3-02 || “corgare | 1-48 |) ° ‘eu5496 | 4-48 | “154504 | 59 
2) 758002 | 3p oisis7 | 1-48. || sasves |. 4-42 "154236 | 58 
g) | 383 | SRE | | Ba) | ie | 
5 | .759492 | 2-9 || “o19999 | 1-47 || “gagsvo | 4-47 153430 | 55 
6 “759672 - 3.00 || 2912883 ae "846839 vin "158161 | 54 
lad < . or, 5 ° ¢ i 
8| ‘toons | 2-98 || ‘ores | 148 || csrare | 4-47 | ctpanot | be 
Ppreronri t+) | 2°00 912566 | 1-48 || ‘eeveaa | 4-48 "159356 | 51 
10 | .760390| $98 || .s1za77| 3-48 || geroia | 4-48 452087 | 50 
11 | 9.760569 9.912388 || 9.g4si8i | 10.151819 | 49 
1g 160748 aa "912299 ot g18449 | 4-47 "151551 | 48 
ef mee | oes | ome | ras | cea) 2 | bee 
15 | ‘761285 | 2-98 || ‘ore0st | 1-52 || “esop54 | 4-47 | “a50746 | 45 
16 | 761464 | 2-28 |P ‘ortose | 1-48 |) ‘sag5a2 | 4-47 | l150478 | 4a 
th| ctorset | 2-98 |) “otis 152 | ‘soosr | 449 | cagoss | 2 
19 | 1761999 | 3-94 911674 | 1-83 || ‘esoge5 | 4-47 "149675 | 41 
pe) stoni77 | 2°3"°.|| corspea | 1-89. | “asonas | 4-48 "449407 | 40 
21 | 9.762356 9.911495 9.850861 |, 10.149139 | 39 
pe) 762534 Bete voitans | 12080 I)” estoy |) 428! "448871 | 38 
ees ee | ces) | eae) | ees 
25 | 763067 | 2-9¢ "911136 | 1-50 "851931 | 4:42 "448069 | 35 
26 | 763245 | 2-90 ‘911046 | 1-20 ‘gs2i99 | 4-4% "147801 | 34 
Pe | ).763428 | 3-0 910956 | 1: ok 952466 | 44> "147534 | 33 
28 | .763600 | 3-3" 910866 | 1-5) || ‘esevaa | 4-45 147267 | 32 
a9) ear? | 2:82 91076 | 4:20 853001 | 4-4 "446999 | 31 
30 | .763954 | 3:32 910686 | 3:2) || .858e68 | 4-4? "146732 | 30 
31 | 9.764131 ~ || 9.910596 9.853585 |, 10.146465 | 29 
a2 | .ve4308 | 3:22. || 910506 | 4-99 || -sssso2| 4-43 | 1146198 | 28 
Bi) “roiee2 | 2-25 |) ‘onoges | 1:90 |) “s5ass6 | 4-43 | “T5664 | 36 
Bye | te | cee | ie) te) ise |e 
er te | we, 28 | ger te | cass 
fi c t .14 
me9 | 765544 ota "909873 a "855671 se "444329 | 21 
540 | 765720 | 3:23 || -oogvse | 3-28 855938 | 4:43 444062 | 20 
41 | 9.765896 9.909691 9.956204 | | 10.143796 | 19 
42 | .766072 ee "909601 ae 856471 Fg "143529 | 18 
| “veetes | 2-28 || “gost | 1-52. || “serons | 4:45. | _“igonee-| 26 
45 | (766598 | 2:92 "909328 | 1-92 ‘g5va70 | 4-48 142730 18 
45 “10077 oe 909237 es "857537 roe 442403 | 14 
766949 | 2-22 ‘909146 | 1-5 "957803 | 4: "442197 | 13 
48 | .vevi2d | 3-32 || :900055 | 4:28 || ‘s5soe9 | 4-43 141931 | 12 
49 | .767300 | 8:23 || ‘o0go64 | 1-52 || ‘eseaze | 4-48 "441664 | 11 
BO | 767475 | 3°95 || 908873 | {28 “958602 | 4-43 "441398 | 10 
bi | 9.767649 : 9.908781 9.858868 | , 10.141132 | 9 
2 | 67st 2-02 || :908690 jie e5gia4 | 4-43 "140866 | 8 
ee) resizes | 2-20 || “Qossor | 283. || cBeegee | 4-43 | ‘iaoma |e 
bs | (768348 | 2-92 ‘g084i6 | 1-52 “g5g932 | 4:43 "140068 | 5 
P56 | -zess22 | 3-2254| 90824 | 7:28 || ‘gs0108 | 4-43 | ‘1s9g02 | 4 
Bpeee ie | elie | wee] te | ee 3 
59 | 769045 | 3-20 908049 | jp || ‘860995 | 4-42 "439005 | 4 
— 60'| 9.769219 | 2: 9.907958 | 2: 9.861261 | +: 10138739 | OF 
125° Cosine. | D. 1”. || Sine. | D. 1”. || Cotang.| D.i". | Tang. |54° 


12 . 





g. Y g 







































































0’| 9.769219 ———— 
1) .769393 | 2-20 9.907958 — 
a| ‘re9566 | 2-88 “oove66 | 1-53 9.861261 5 = 
3 ~e9740 | 2 90 907774 1.58 861527 4.48 0.188739 | 60° 
4| ‘469913 | 2-88 ‘907682 | 1-58 'g6i7g92 | 4:42 .138473 | 59 
5! ‘770087 | 2-90 ‘gov590 | 1-58 "gg2058 | 4:48 .138208 | 58 
6 rm) 960 2 88 907498 153 862323 4,42 137942 5Y 
| ‘wrq433 | 2-88 ‘gov406 | 1-58 862589 | 4-43 131607 | 56 
tlie | Sahl cheese eet |) Seeebee Ih Gian 137411 | 55 
9| “e077 2 88 “907222 1.53 "e63119 | 4: 42 .137146 | 54 
10 | ‘770952 | 2-88 ‘ga7i29 | 1-58 963385 | 4:23 136881 | 53 
11 | 9.771 2.88 ‘govos7 | 1-53 "g63650 | 4:42 .136615 | 52 
1 | > -anBog 9.90 Pe3 | ||| 86806: |!) Fare 136350 | 51 
13 771298 2.88 . 906945 4,42 136085 | 50 
ii ‘wrta7o | 2-87 “goea5e | 1-55 9.864180 10 
74643 | 2-88 "906760 | 1-53 "ge4445 | 4:42 .135820 | 49 
15'| °.774 2.87 “906667 | 1:55 "964710 | 4:42 135555 
815 48 
“ ‘ewr9g7 | 2-87 90657 1.53 "gg4975 | 4:42 135290 | 47 
is “nvaipg | 2-87 "906482 | 1-55 865240 | 4.48 135025 | 46 
. “779331 2 87 -906389 1.55 .865505 4.42 184760 | 45 
: “wagng3 | 2.8% “9062906 | 1-29 86577 4.42 134495 | 44 
“ners | 2-87 “906204 | 1-53 "966035 | 4:48 134230 | 43 
91 | 9.7’ 2.87 ‘906111 | 1-22 "g66300 | 4:42 133965 | 42 
172847 1 86 4.4 1337 
a3 | ee 9.9060 "a5 || | 866564 |g 0 00 | 41 
Sek i 773018 2.85 : 18 4.42 133436 | 4 
31 agg | 2-87 ‘905925 | 1-55 9.866829 40 0 
9 “wvgge1 | 2-85 "905832 | 1-58 "gev004 | 4-42 .183171 | 39 
25 | 778583 2-8 || | BOBTBT |. ape “poaee |i. 400” |) eerste 
oy wegnog | 2-89 "gon645 | 1-36 "gevgo3 | 4-42 .132642 | 37 
a ‘ongars | 2.85 ‘905552 | 1:59 "geveey | 4:42 .182377 | 36 
29 | rai 2.85 || 905866 | 7"p> eee |. 4a08 | edge fae 
0 | “rrageg | 2-85 “gon272 | 1-57 “geseso | 4-40 131584 | 33 
$i | 6p 283 ‘oo5i79 | 1-32 g6s045 | 4:42 . 131820 | 32 
> | aioe 558 9 1.57 69209 | 4:40 131055 | 31 
a “aagvag | 2-85 . 905085 4.40 .180791 
33) erasoo 563. || -904992 iss || 9-869473 | 4 30 
34) 715010 2-83" || 1903898 | t'py BoeeS |) aoe | Pe ee 
"775240 2.83 904804 1.57 870001 4.40 . 180263 | 28 
36 | 775410 9193 || -904711 155 || -820265 4.40 .129999 | 27 
os " wes50 2 83 904617 1.57 "g70529 | 4-40 129735 | 26 
: “wasmng | 2: 83 904523 1.57 "870793 | 4: 40 129471 | 25 
. “wwsgon | 2-83 "904409 | 1-97 ‘s71057 | 4:40 . 129207 | 24 
Hayes || 2083) ||) abaede Ler | Saetaed || dogs 128943 | 28 
ai | 9% 292 “904041 | 1-52 ‘971585 | 4:40 .128679 | 22 
5 776259 1.57 ‘grigag | 4-40 128415 
: WBA 99 2 83 9.904147 4.38 128151 pe 
43 | 770598 2:88) ||, poms | f'pe 9.s72iie | * 
44 | “ra0t0s 2-82 || “oososa | 1-28 |) | eae | agg” | SCA ae 
5 “wyeon7 | 2-82 “oo3864 | 1-58 ‘g72¢40 | 4-40 127624 | 18 
46 | 777106 2B « |) Soommo |) Tage Bee | 488° | tater (te 
is “wamoan | 2.82 “903676 | 1 57 "eraig7 | 4-40 127097 | 16 
: “wenga4 | 2-82 “go35g1 | 1-58 "g73430 | 4-38 126833 | 15 
. “wenegig | 2-82 "903487 | 1-27 "g73694 | 4-40 126570 | 14 
‘wanmgy | 2-80 “903392 | 2-58 "973057 | 4.38 .126306 | 13 
bi | 9.777 2 4]! Teosens |) Hine ‘rise, | 488: | : eee ae 
59 777950 1.58 'g74484 | 4-40 . 125780 
9 | “regi | 2-82 9.903203 4.38 125516 is 
53 | 8287 3/99 || 903108 158 || 9.874747 | 4 10 
4 ‘nmggae | 2-80 “903014 | 1-32 ‘ge5010 | 4-88 10.125253 | 9 
a | heer 2 ||) Soeeee bape it). g75e73 | 4-38 -124990 | 8 
- "wagegg | 2-80 “goa8e4 | 1-58 “grapgy | 4-40 "424797 | 7 
BY | .778960 oes iit cotieted 1! 7 ee See) 4 ee oe 
8 | 779128 DS 976063 | 4:38 124200 | 5 
Pl eae 2D ||) sods | 1:28 Brees | 4a aes oe 
9 weg4g3 | 2-80 “902444 | 1.58 “ere5sg | 4:38 123674 | 3 
; ieee _—— 9.902349 1.58 876852 4.38 128411 | 2 
26° Cosine. | D. 1” ents 9.877114 4.37 . 123148 1 
fede Sine. re 10.122886 0' 
- pee Cotang. D 1" —— |-——- 
oS ae Tang. |53° 





9.779463 





COSINES, TANGENTS, AND COTANGENTS 


Sine. 


781134 
781301 
. 781468 
. (81634 
(81800 
781966 
- (B2182 
782298 


(82464 | 


.(82630 
(82796 
. 782961 
(83127 


“785925 
“786089 


- (86252 
(86416 
786579 
186742 
. 786906 
787069 
(87232 
. (87395 
T87557 
787720 
. (87883 
7188045 
. 788208 
. (88370 
~ (88582 
. 788694 
. (88856 
. 789018 


789180 |. 


9.789342 


127° Cosine. ie Ge 
er 


09 09 WW WN WW WWWWNWWWD 











~ WNWWWNWWWWWI WNW WNVNVWWWW WHWwNwy 











Cosine, 


9.902349 
902253 
902158 
902063 
901967 
901872 
90177 
.901681 
901585 
.901490 
901394 


9.901298 
-901202 
.901106 
.901010 
. 900914 
. 900818 
. 900722 
- 900626 
900529 
.900433 


9.900337 
900240 
.900144 
900047 
899951 
899854 
899757 
899660 
899564 
899467 


9.899370 
899273 
899176 
.899078 
.898981 
«898884 
898787 
.898689 
«898592 

898494 


9.898397 
898299 
898202 
898104 
.898006 
.897908 
897810 
897712 
897614 
897516 


9.897418 


Sine, 



































Tang. 


9.877114 


877377 
877640 
877903 
878165 
878428 
878691 
878953 
879216 
879478 
879741 
9.880003 
880265 
880528 
. 880790 
881052 
881314 
881577 
881839 
882101 
882863 


9.882625 
882887 
.883148 
883410 
888672 
883934 
884196 
884457 
884719 
. 884980 


9.885242 
- 885504 
885765 
886026 
. 886288 
886549 
886811 
887072 
887333 
887594 


9.887855 
. 888116 


9.890465 
890725 
890986 
891247 
891507 
891765 
892028 
892289 
892549 

9.892810 








D. 1 


qo co O29 G9 C9 C2 09 
HAIQVIVIAS 


4,37 


a Ss 
weeweweoet owe cw oD< 
RSASRERAS RAH 


Cotang. | D. 1". 





| 
Cotang. 142° 


10122886 
122623 
. 122360 
. 122097 
. 121835 
121572 
1213809 
121047 
120784 
120522 
~ 120259 


10.119997 
119735 
119472 
119210 
. 118948 
. 118686 
118423 
118161 
117899 
117637 

10.117375 
117118 
116852 
116590 
116328 
116066 
115804 
115543 
11528f 
115020 


10.114758 
114496 
114235 
118974 
118712 
118451 
118189 
112928 
112667 
112406 


10.112145 
111884 
111622 
111361 
111100 
110839 
110579 
110818 
110057 
109796 


10. 109535 
109275 
109014 
108753 
108493 
108232 
107972 
107711 
107451 


10.107190 
Tang. {62° 





127 





SHWww RAI IDO 





128 TABLE XXV.—LOGARITHMIC SINES 


* 






































| 

38° Sine. D.:1".,]| (Cosine, |. |Dod’. | Tang. | D. 1’. | Cotang. 141° 
0’| 9.789342 wn || 9.896532 9.892810 10.107190 | 60’ 
1| 789504 See "896433 ie "893070 aes "106930 | 59 
2| \7go665 | 3:65 || .896835 | tgs 893331 | 4-3 "406669 | 58 
3| cvsgse7| 2:9 || 1996236] J:pe g03591 | 4-33 "106409 | 57 
‘| | oe | eee te | ae te | ele 
6 | _790310 es 895989 Le 994372 | 4-38 105628 | 54 
7 | ‘rgoa71 | 2: "g95840 | 2- "394632 | 4: 105368 | 5 
8 | 790632 2a "895741 ie ‘| g94899 ae "405108 | 52 
9 | 1790793 | 3°68 || ‘sone4i | 1-62 895152 | 4-23 "104848 | 51 
10 | 790954 | 2°68 || ‘sonsae | 1-65 || ‘sosaie | 4-53 404588 | 50 
11 | 9.791115 | 5 9.895443 | 9.895672 10.104328 | 49 
12| (791975 | 2-8% || ““gg5g43 | 1.87 "g95932 | 4:33 104068 | 48 
13 | .791436 | °2-88 go5ed4 | 1-65 goeig2 | 4-38 103808 | 47 
14| 1791596 | 2:82 || ‘go5i45 | 1-65 || “gog45e | 4-38 103548 | 46 
15 | ‘vo1757 ae "895045 te | “996712 is "403288 | 45 
Bee) oor) BRE) Pe | weet) fe | en 
18 | ‘v92937 | 2-87 ‘goiz46 | 1.6% "g97491 | 4:38 "402509 | 42 

etek 2 dae Monte 1.67 Caweey | 4.00 Eat 
19 | :7oz307 | 3:67 || ‘soseas | 1-8 poriolal it te 402249 | 41 
20 | <vozss7 | 3:62 || ‘eoa546 | 1:67 || “soso10 | 4:38 "401990 | 40 
21 | 9.792716 9.894446 '| 9.898270 10.101730 | 39 
22 | 799876 2-67 || 894346 1 808530 | 4-33 | “101470 | 38 

lod =< . ADAR . lad -t 
eee eee ae eae 
25 | ‘793354 | 2-65 "g94046 | 1-6% "g99g08 | | 4:32 "100692 | 35 — 
26 | (793514 at "893946 te 899568 nee "400432 | 34 

ene . 9 ‘ toa of 6 ‘ 

2 | vrossss | 2-5 |) caus] 1-68 | ‘oooowr | 423 | cono13 | 32 
29 | iro3991 | 3:69 993645 | 1-67 || ‘900346 | 4°32 099654 | 31 
30 | 1794150 | 2-65 |) “eoasaa | 3-68 900605 | 4°33 “099395 | 30 
31 | 9.794308 | 5 gs || 9.893444 | 4 6, |] 9.900864] 4 35 | 10.099136 | 29 
32 | 794467 | 2-8 "g93343 | I: ‘901124 | 4: "098876 | 28 
33.| 794626 | 2-6 g9s243 | 1-67 901383 | 4-32 098617 | 27 
31 | 1794784 are "993142 Le | Contes) fee "098358 | 26 
Baie) 2G | Bae) i | S| ee | aap |S 
37 | “y9pe59 | 2-63 "ggag39 | 1.68 "902420 | 4:38 "097580 | 23 
38 | 795417 ee 399739 tee "902679 ates "097321 | 22 
39 | 795575 | 3°83 892638 | 4-68 || ‘oog038 | 4°28 "097062 | 21 
40 | .795733 | 3°63 892536 | 1-70 903197 | 4°32 "096803 | 20 
41 | 9.795801 | 5 g9 || 9.892485 | 4 9g || 9.903456 | 4) | 10.096544 | 19 
42 | .796049 | 3-63 “992334 | 1-68 ‘903714 |, 4-30 "096286 | 18 

noe. 0K 992° 2 ny ae oa 
B) Tae is | See) i | Beka) das | ae | 
45 | 796521 | 2-82 || “g9a030 | 1-7 "904491 | 4:32 "095509 | 15 
46 | 796679 se "391929 cae "904750 a 095250 14 
47 | 1796836 | 2-62 || ‘soige7 | 2-70 905008 | 4°30 3 
48 | .796993 | 5°95 || .so1726 | 1-6 905267 | 4°35 094733 | 12 
50 | 797307] 5-05 || .goines | 3-85 || “oos7es | 4-38 "094215 | 10 
B1 | 9.797464 | 9 go || 9.891421 | 4 no || 9.906043 | 4 99 | 10.098957 | 9 
Bo aetaviel || sew] uanela dt any 906302 | 4-38 "093698 | 8 
eae | ce | cme as | men Bae 
55 | 798091 | 2-62 "g91013 | 2-4 ‘govo77 | 4-30 "092923 | 5 
56 | “roget7 | 2-60 |) ‘googr1 | 2-7 907336 | 4-32 ‘092664 | 4 
57 | "798103 | 2-69 |] “goog09 | 1-70 ‘907594 | 4-30 ‘092406 | 3 
58 | |v98560 2-63 || ‘890707 oy "907853 mee "092147 | 2 
59 | ivosvie | 3:69 || ‘go0605 | }-¥ S0BLIL Al agen "091889 | 1 
60’| 9.798872 : 9.890503 af 9.908369 ? 10.091631 | 0” 


























128° Cosine. | D. 1". Sine. D, 1". || Cotang. “Do 1". Tang. /51° 








¢ 


COSINES, TANGENTS, AND COTANGENTS 











39° Sine. 
0’| 9.798872 
1 - 799028 
2 . 799184 
3 .799339 
4 ~799495 
5 -799651 
6 . 799806 
v¢ . 799962 
8 .800117 
9 . 800272 
10 800427 
11 | 9.800582 
12 .800737 
13 |, .800892 
14 .801047 
15 .801201 
16 .801356 
17.| .801511 
18 .801665 |. 
19 .801819 
20 .801973 
21 | 9.802128 
22 . 802282 
23 802436 
24 802589 
25 .802'743 
26 .802897 
27 | .803050 
28 . 808204 
29 803357 
30 .803511 
31 | 9.803664 
32 .808817 
83 .80397 
84 -804123 
35 .80427 
36 .804428 
37 .804581 
38 . 804734 
39 804885 
40 . 805039 
41 | 9.805191 
42 .805343 
43 805495 
44 .805647 
45 805799 
46 ~805951 
47 .806103 
48 . 806254 
49 .806406 
50 .806557 
51 | 9.806709 
52 . 806860 
53 .807011 
54 .807163 
55 .807314 
56 807465 
57 .807615 
58 | .807766 
59 .807917 
60’| 9.808067 
129° Cosine. 


WWIWWWWW WWWWWWNWNWWIO WWIwwi 


4 OED bes 


=> 
i) 


OVOTOTOTOUOTOU OTOLOTOTOTOTOUOUOT ON OT OT OT OWS O'S> Sot 
KRIAKIIHRI DRIVIVHDRDDD ZeOBSHSSERS 


SIC SSIS Over 
WWSHWWHWMW WW 





D. ee 




















129 





Cosine. 


9.890503 
890400 
-890298 
890195 
890093 
889990 
889888 
889785 
. 889682 
889579 
889477 


9.889374 
889271 
889168 
889064 
. 888961 
888858 
888755 
888651 
.888548 
888444 


9.888341 
888237 
888134 
888030 
887926 
887822 
887718 
887614 
887510 
887406 


9.887302 
887198 
. 887093 
886989 
886885 
886780 
886676 
886571 
886466 
886362 


9 .886257- 
886152 
. 886047 
885942 
885837 
885732 
885627 
885522 
885416 
885311 


9.885205 























Di i: Tang. 
wo || 9.908369 
1.72 || 908628 
1.70 || {908886 
tA 909144 
119 || ‘909402 
ig "909660 
10 || 1909918 
is 910177 
as 910435, 
tay || 910693 
ting |] - 910951 
9.911209 
1.7 || 911467 
to 911725 
1.73 || 1911982 
1.73 || 1912240 
iid || - 912498 
rie 912756 
13 || (918014 
vores th .9u8e7t 
1.73. || 9-918787 
1.73 || ‘914044 
re 914302 
1.73 || 1914560 
1.2 || 1914817 
1-2 || ‘915075 
173 || 1915382 
ne 915590 
ee 915847 
1/3 || ‘o1e104 
wa || 9.916362 
PM 916619 
ee "916877 
17° || cor7i34 
ie 917391 
Hi 917648 
1.73 || ‘917906 
ae 918163 
tn "918420 
iO 918677 
9.918934 
i "919191 
i 919448 
ie "919705 
al "919962 
ee 920219 
1h 920476 
11> || 1920738 
ie "920990 
ne "921247 
9.921503 
ye "921760 
ae 929017 
17 || 920074 
1 |] 1922580 
ie "923044 
i "923300 
te || lopass7 
9.923814 











Di ae 


a 
oo co Go Go GO CO GO OD 
Swooooeow 


wwne PO ee 
DAD DODDS MO® 


4.28 


D.1”. || Cotang. | D. 1". 


| 
Cotang. 140° 


10.091631 
091372 
-091114 
090856 
.090598 
.090340 
.090082 
089823 
089565 
.089807 
089049 


10.088791 
088533 
088275 
088018 
087760 
087502 
087244 
086986 
086729 
086471 


10.086213 





.085956 | 


.085698 
085440 
085183 
084925 
.084668 
084410 
084153 
083896 


10.083638 
083381 
088123 
.082866 
082609 
082352 
082094 
.081837 
.081580 
081823 


10.081066 
080809 
080552 
-080295 
.080038 
079781 
079524 
079267 
.079010 
078753 


10.078497 
078240 
077983 
077726 
077470 
077218 
076956 
076700 
076443 

10.076186 





Tang. 














130 TABLE XXV.—LOGARITHMIC SINES 









































40° Sine. Die Cosine. 
0’) 9.808067 | 9 no || 9.884254 
3 | “Bosses | 2-50. || “eeaoss 
3| ‘sog519 | 2:22 "883936 
"808669 | 2°20 || “gssga9 
. “808819 2.50 883723 
6 | :808969 | 2:°0 || '383617 
250 - 8 
7 | .809119 | 3:3) 883510 
8| .309269 ) 3:95 883404 
9 | 909419 | 8:25 883297 
10 | .809569 | 3°33 883191 
11 | 9.809718 | 9 x9 || 9.883084 
12 | 809868 | 3-2 882977 
13 | ‘810017 | $56 882871 
14| .si0167 | 5°93 882764 
35 | :810316 | 3-43 882657 
16 | .810465 | 3-48 882550. 
17 | -si0614 | 3-43 882443 
18 | -810763 | $48 "882336 
19 | .sio912 | 3:4 "882229 
20| .8i1061 | 3:48 882121 
21 | 9.811210} 4 yy || 9.892014 
92 | 811308 | §-45 881907 
24| .sii6s5 | 3-44 881692 
| sie | aa || “aie 
247 joie 
27 | .812100 | S46 88136 
28 | .sieas | 3:4¢ || .881261 
29 | .812306 | 344 "881153 
30 | 1812544 5:45 || 881046 
31 | 9.812692 | 5 47 || 9.880938 
a3 | tages | 2:47 || ‘gepres 
= ee 2.45 (eae 
24 ‘813135 | 3'4 "880613 
35 | 813983 | 5-42 “880505 
36 $1380 Say || .880307 
( , 578 2 45 . 880289 
38 | 1813725 | 3-42 880180 
39 | 813872 | 3°4e || -s9o072 
40 | 1814019 | 5-4? || ‘879963 
41 | 9.814166 | 9 4x || 9.879855 
42 | .814313 | 3:42 879746 
Ao ae 
45 | 1814753 | 2-43 || ‘979490 
46 | .814900| 242 || “evo3i4 
47.| '815046 | 2-43 || ‘g79902 
48 | 815193 | 2-43 ||  “gr9093 
ae 2.43 proasG 
B1 | 9.815632 | » 45 || 9.878766 
B2 | 815778 | 5°45 || -878656 
s5 | ‘sianis | 243 || sms 
56 | 816361 | 2: 87821 
See? || boas. || haeeaee 
bo | “B1e768 < “377890 
60’| 9.816943 | 9.877780 
130° Cosine. | D. 1" Sine. 











D. 1’. 


@ 
o 











Tang. 


9.926634 
. 926890 
927147 
927408 
927659 
927915 
928171 
928427 
. 928684 
. 928940 


9.929196 
929452 
929708 
929964 
930220 
930475 
930731 
. 930987 
931243 
-931499 


9.931755 
-932010 
932266 
9382522 
-932778 
. 938033 
938289 
- 938545 


“936611 


9.936866 
. 937121 
937377 


9.939163 





Cotang. 





} 
Cotang. 139° 


o 
— 


WWW WNYWWNVNVNVNWWHYDYD KYONYNWWVWHWNYVDYD VWNNWNWVYHWN WWD WNW WKY NWNWNWW NWWNWNNWNNHwNNYD 
RRRNVRRRSR RKSRSRRRNNN RVRN % 


VIBAIVIBDRHA 


VIIBDIQIANHS A 


VHIIAHT VIVA SAA AAS 











10.076186 | 60/ 
075930 | 59 
075673 | 58 
075417 | 57 
075160 | 56 - 
074904 | 55 
074648 | 54 
074391 | 53 
074135 | 52 
073878 | 51 
073622 | 50 

10.073366 |-49 
073110 | 48 
.072853.| 47 
072597" | 46 
072341 | 45 
072085 | 44 
071829 | 43 
071573 | 42 
071316 | 41 
071060 | 40 


10.070804 | 39 
.070548 | 38 
.070292 | 37 - 
.070036 | 36 
.069780 | 35 
.069525 | 34 
.069269 | 33 
.069013 | 32 
-068757 | 31 
.068501 | 30 


10. 068245 | 29 
067990 | 28 
067734 
067478 | 26 
067222 | 25 
066967 | 24- 
(066711 | 28 
066455 | 22 
066200 | 21 
065944 | 20 


10.065689 | 19 
.065433 | 18 
.065178 | 17 
.064922 | 16 
.064667 | 15 
.064411 | 14 
.064156 | 13 
.063900 | 12 
.063645 | 11 
.063389 | 10 

10.063134 | 9 
062879 
-062623 | 7 
.062368 | 6 
.0621138 | 5 

.061858 | 4 

3 
2 
1 





E 


-061602 
.061347 
. 061092 
10.060837 | 0’ 





COSINES, TANGENTS, AND COTANGENTS 13] 

















41°; Sine. 
0 | 9.816943 
1 .817088 
2 .817233 
3 .817379 
4 .817524 
5 817668 | 
6 .817813 
¢ .817958 
8 .818103 
9 .818247 
10 .818392 
11 | 9.818536 
12 .818681 
13 .818825 
14 .818969 
15 .819113 
16 .819257 
ey .819401 
18 .819545 | 
19 .819689 
20 .819832 
21 | 9.819976 
22 .820120 
23 . 820263 
24 .820406 
25 .820550 
26 .820693 
RC . 820836 
28 .820979 
29 .821122 
80 .821265 
81 | 9.821407 
32 . 821550 
33 .821693 
34 .821835 
35 .821977 
36 . 822120 
37 822262 
38 , 822404 
39 822546 
40 .822688 
41 | 9.822830 
42 |. .822972 
43 823114 
44 823255 
45 .823397 
46 823539 
47 823680 
48 823821 
49 823963 
50 824104 
51 | 9.824245 
52 .824386 
53 -824527 
54 824668 
55 .824808 
56 , 824949 
57 . 825090 
58 825230 
59 82537 
60/| 9.825511 
131° Cosine. 


{WA WWW I WW WW WWW WWWIWI WWWW WI WWII WWWIWWWWWWI WWWIWWWWWII WIWWIWIWIOWIWIWIDW 


D, I. 


ee 


Bwwk Pw LLL LEER PRE 


© C eeuN WN to WNW WWWwWWNw Www > ee 


1 

















Cosine. 


9.877780 


877670 
877560 
877450 
877340 
877230 
877120 


876789 
876678 
9.876568 
876457 
876347 
876236 
876125 
876014 
875904 
-875793 
875682 
875571 


9.875459 
875348 
875237 
875126 
875014 
874903 
874791 
.874680 
874568 
874456 


9.874344 
.874232 
.874121 


- 1873335 
9. 873228 
'873110 
872998 
{872885 
‘872772 
872659 
(872547 
1872434 
1872321 
872208 


9.872095 
871981 
-871868 
871755 
871641 
871528 
(871414 
871301 
871187 

9.871073 





Sine. 











D. 1’. 





_ 
(es) 
oo 


PL peek ek pk ek ek ek fk pk ek ek ek ek fk peek pk ek peeked fk Rk ek a pe pk fo pak fk pak ek ek fee pk fk eek fk ek kek ek ek pak kk ek ek ke pe et tk 
[e2) 
oo 


SOS Msewomwm DOMOMWOMOMOD 


3 
i 





Tang. 


9.939163 
939418 
939673 
939928 
940183 
940439 
940694 

* .940949 
941204 
-941459 
-941713 


9.941968 
942223 
942478 
942733 
942988 
943243 
948498 
943752 
944007 
944262 


9.944517 
94477 
945026 
945281 
945535 
945790 
946045 
946299 
946554 
.946808 


9.947063 
947318 
947572 
947827 
948081 
948335 
948590 
948844 
. 949099 
949353 


9.949608 
949862 
950116 
950371 
950625 
950879 
951133 
951388 
951642 
951896 


9.952150 
952405 
952659 
952918 
953167 
~ 953421 
953675 
953929 
7954163 

9.954437 








Cotang. 





Dear. 


2 2 29 29 2% 
RRRFSRTRE 


pA AAp AAP AR A 
02 0D 
Ove 


Q ) 0 0 29 ~ 
oor ew) $2.29 £9 29:29 29 29.09.29 29 oto ewer 


@ WNWNWNWWONWNWNWYW 


WWW NWNW*W 


4". 


10.060837 | 60’ 








| 
Cotang. 138° 


:060582 | 59 
060327 | 58" 
060072 | 57 
059817 | 56 
(059561 | 55 
059306 | 54 
"059051 | 53 
:058796 | 52 
“058541 | 51 
058287 | 50 


10.058032 | 49 
057777 | 48 
1057522 | 47 
057267 | 46 
1057012 | 45 
056757 | 44 
(056502 | 43 
1056248 | 42 
(055993 | 41 
055738 | °40 

10.055483 39 
055229 | 38 
1054974 | 37 
054719 | 36 
1054465 | 35 
(054210 | 34 
053955 | 33 
058701 | 32 
(058446 | 31 
[053192 | 30 

10.052987 | 29 
[052682 | 98 
052428 | 97 
1052173 | 26 
1051919 | 25 
1051665 | 24 
1051410 | 93 
‘051156 | 22 
"050901 | 21 
050647 | 20 


10.050392 | 19 
.050138 | 18 
.049884 | 17 
.049629 | 16 
-049875 | 15 
.049121 | 14 
.048867 | 13 
.048612 | 12 
-048358 | 11 
.048104 | 10 

10.047850 
047595 


"045817 
10045563 


Tang. in 


~ 





S 
P= 
fer) 
Or 
3 
ie) 
| Ce OW ROTONWOO 


oo 
° 





132 











42°, Sine. 
0’| 9.825511 
1 .825651 
2 .825791 
3 .825931 
4 .826071 
5 826211 
6 .826351 
ui .826491 
8 826631 
9 .826770 
10 .826910 
11 | 9.827049 
12 827189 
a 827328 
14 827467 
15 .827606 
16 827745 
17 .827884 
18 828023 
19 -828162 
20 828301 
21 | 9.828489 
De . 828578 
eo .828716 
24 . 828855 
25 .828993 
26 .829131 
fi . 829269 
28 829407 
29 . 829545 
30 .829683 
31 | 9.829821 
32 .829959 
33 . 830097 
34 830284 
35 .830372 
36 .830509 
Si 830646 
38 830784 
39 .830921 
40 .831058 
41 | 9.881195 
42 . 831332 
43 . 831469 
44 .831606 
45 .831742 
4G .831879 
47 . 832015 
48 . 882152 
49 .8382288 
50 . 8382425 
5i | 9°882561 
2 832697 
53 .832833 
54 832969 
55 833105 
56 .833241 
57 833377 
58 .833512 


59 | .833648 
60’| 9.833783 


132° Cosine. 




































































TABLE XXV.—LOGARITHMIC SINES 
r 
Drift Cosine. | D. 1". Tang. D. 1’. | Cotang. 137° 
9.871073 9.954437 oa | 10.045563 | 60’ 
2-33 || 1870960 ee 954601 | 4°33 045309 | 59 
aie 870846 | 7-29 954946 | 4-33 045054 | 58 
Sees || pes@e0732 | G°a, “) qrseomen0 IP gine "044800 | 57 
ge 870618 | 3-29 955454 | 4°23 "044546 | 56 
ae 870504 | 1-39 955708 | 4°33 "044292 | 55 
aie 870300 | 3-30 955061 | 4-32 "044039 | 54 
ates 870276 | 1-39 956215 |. 4°38 043785 | 58 
ges 870161 | 7°23 956469 | 4°32 043531 | 52 
Bice 870047 | 3-20 956723 | 4°53 | ,.048277 | 51 
aS | ee] 18 | oe | 28 | oe 
9572: 0427 
gree "869704 re “957485 rie "042515 | 48 
pee 869589 | 1-22 957739 | 4:38 "042261 | 47 
Bae geod74 | 1-98 957993 | 4-23 "042007 | 46 
eee 369360 | 1-90 958247 | 4-33 (041753 | 45 
2-32 || ‘gege4s | 1-98 958500 | 4°53 "041500 | 44 
Coe ll qBeats0\| eave 958754 | 4°58 "041246 | 43 
g:82 ll s880015 || 4°92 959008 | 4°33 "040992 | 42 
pine 868900 | 4-22 7959262 | 4°53 "040738 | 41 
iS | (eee | 8 | eee | 8 | al 
86867 9597 
2-32 || .868555 192-1] 1960023 neh "039977 | 38 
2-30 ||, 868440 | 4-92 960277 | 4-23 039723 | 37 
see 968324 | 1-93 960530 | 4°38 039470 | 36 
ep g6g209 | 1-92 960784 | 4:33 039216 | 35 
be. 868093 | 7-23 961088 | 4°33 038962 | 34 
Bee Hl agneiais | ates 961292 | 4°33 038768 | 33 
eo 967362 | 4-93 961545 | 4:23 038455 | 32 
ech 867747 | 1-82 961799 | 4°55 038201 | 31 
a8 | Sei) 1 | ec | ae 
9.867515 9962306 
ed "867399 te "962560 ve ‘037440 | 28 
oo. 867283 | 4-33 962813 | 4°38 ‘O37187 | 27 
aa 867167 | 1-23 963067 | 4-33 "036933 | 26 
eee 867051 | 4-83 963320 | 4-32 "036680 | 25 
Bo “866935 | 1-73 963574 | 4-28 "036426 | 24 
ae 866703 | 1-23 964081 | 4-33 "035919 | 22 
2-28 || ‘sense | 1-88 964885 | 4°38 "035665 | 21 
oS | ee] 18 | es] 18 | ee 
86 9.964842 ; 
es 966237 | 3:93 || “‘965095 | 4-22 "034905 | 18 
oon 866120 | 4-95 965349 | 4°33 "034651 | 17 
eae 866004 | 7-23 965602 | 4-23 “034398 | 16 
pooh 865887 | 1-29 965855 | 4°33 "034145 | 15 
ae 865770 | 1-38 966109 | 4°33 "033891 | 14 
bok "865653 | 1-99 966362 | 4-38 "033638 | 13 
oa 7865536 | 1-88 966616 | 4°33 “033384 | 12 
eae 865419 | 1-35 "966869 | 4-33 033131 | 11 
oe “865802 | 1-38 967123 | 4°33 "032877 | 10 
9.865185 9.967376 10.032624 | 9 
eet 865068 ae 967620 | 4:22 | 032371 | 8 
att 864950 | 4:34 ‘967883 | 4°38 (032117 | 7 
od 864833 | 1-98 968136 | 4°35 | -081864 | 6 
9 97 864716 | 3 "on -968389 | 4°93 .0381611 | 5 
Siep s64508 | 1-37 968643 | 4-33 "031357 | 4 
aie g644gi | 1-95 968896 | 4°33 031104 | 3 
fis 864363 | 4-97 969140 | 4-32 “030851 | 2 
as || peeeete dy tebe "969403 | 4°33 -030597 | 1 
9.864197 | 1: 9.969656 | +: 10.030344 | 0” 
D1", {| Sine,- | D. 1°. || Cotang. | D.1"%; | Tang. (47° 





COSINES, TANGENTS, AND COTANGENTS 





43° Sine. 
0’| 9.883783 
1 . 838919 
2 . 834054 
3 .884189 
4 834325 
5 834460 
6 .834595" 
vi .834730 
8 | .834865 
9 .834999 
10 835134 
11 | 9.835269 
12 . 835403 
13 . 8855388 
14 . 8385672 
15 835807 
16 835941 
17 836075 
18 836209 
19 836343 
20 836477 
21 | 9.836611 
22 . 8386745 
23 .836878 
24 .8387012 
25 837146 

26 83727 
27 .887412 
28 | .837546 
29 .837679 
30 837812 
31 | 9.887945 
32 .888078 
33 838211 
34 | .838344 
35 | .838477 
36 .838610 
37 | .888742 
38 | .838875 
39 .839007 
40 | .839140 
41 | 9.839272 
42 839404 
43 | 839536 
44 | 839668 
45 | .839800 
46 .839932 
47 | .840064 
48 .840196 

49 84032: 
50 .840459 
51 | 9.840591 
52 .840722 
53 | .840854 
54 |} .840985 
55 .841116 
56 -841247 
57 .841378 
58 .841509 
59 841640 
60’ | 9.841771 


133° Cosine. 


4 





Dol 


WNWNWNMNVNHWWNWNWYW 


WNWWWW 


DoD HWNWWW 


WNMWNWNWwNwKNW ) 
& ERSSSRSSSES SSSSSERRRS OULU) OL OT OT OU OT OT HE 


5 
i 


WNMWWNMWNWWe 


WWWWWWNWW WWHWWNYVWVNNY WHHYNWVWWVNYNWY WHWNMNVWNWWVNNY WHWNWWWNWVUNVNY WWMM WWIOWW WD 
COWOWOCKNVWWNWW 


Ree DH WH WWW WNWONWYDW 
OO G0 GD CO BY GO GO © OO Sa>SSSSSSSE8 























Cosine. 


9.864127 
.864010 


"862946 


9.862827 
862709 
.862590 
862471 
862353 
862234 
862115 
861996 
.861877 
861758 


9.861638 
.861519 
.861400 
-861280 
.861161 
.861041 
860922 
. 860802 
.860682 
. 860562 


9.860442 
860822 
860202 
860082 
859962 
859842 
859721 
859601 
859480 
859360 


9.859239 
859119 
858998 
858877 
858756 
858635 
858514 
858393 
858272 
858151 


9.858029 
857908 
857786 
857665 
857543 
857422 
857300 
857178 
857056 

9.856934 





Sine. 


DT 


rare 
ve) 
OU 





.98 


WWWW MWR DH HB We HHH HEHE REP Pi Pi Pt Pt Pt Pt Pt Pt Pe et 
ive) 
(ee) 


SOoSofooo Sooo DOoSs 


WW WW WW WWwwDw 
i SD 6 
SSLSBSVSS SUESSTRREHS SSSESSSSSES SSSSKRSRSRE SF 


ofS 


a) 


WHMWWWNWNWW WWWWWWWWWDY 
oooocoo 


ooorlocecoce&o 














Tanga De ks 


9.969656 
969909 
970162 
970416 
970669 
970922 
971175 
971429 
971682 
(971935 
972188 

9.972441 
9712695 
972948 
973201 
973454 
973707 
973960 
974213 
974466 
974720 

9.974973 | 
975226 
975479 
975732 
975985 
976238 
976491 
976744 
976997 
977250 

9.977508 


WNWNWNWNNWWWY 
WWWMW WOW 


rw) 


t ~} WW WW 
VHHVYVHVS BWNWNW 





i 
rw) 











9.980033 
. 980286 
. 9805388 
980791 
981044 
. 981297 
981550 
.981803 
. 982056 
. 982309 


9.982562 
. 982814 
983067 
983320 
983573 
. 983826 
. 984079 
. 984332 
. 984584 

9.984837 


“Cotang. | D. 1". 


S VONWUWVMNWNONUWSCHO HONWONWOWOWNWWOWN WWW 


WwWNwWvwvwvww 2 v9 WNWNWNWWwWNW WWW 
VEVVeevve WNWNWNWNHMONMNWNAWN HWNWNWW < 


AAR ADL AAR A AA DADRA AA RADAR AA RA BBB, 











133 


i 
Cotang. 136° 


10.030344 
030091 
029838 
029584 
029331 
“029078 
028825 
“028571 
028318 | 
028065 
027812 

10.027559 
027205 
“027052 
026799 
026546 
026293 
026040 
025787 
025534 
025280 

10. 025027 
02477 
(024521 | 
024268 
"024015 
023762 
"023509 
"023256 
023003 | 
“022750 


10.022497 
022244 
.021991 
021788 
-021485 | 
021232 
020979 
-020726 
020473 
020220 


10.019967 
019714 
.019462 
.019209 
018956 
018703 
.018450 
.018197 
017944 
017691 


10.017438 
.017186 
0169383 
.016680 
016427 
016174 
.015921 
015668 
015416 

10.015163 


Tang. |46° 


or 
~t 


or 
Ne 





s 


co 
lor) 


Qe dW COR OT) FO 





134 








44° Sine. 
0’| 9.841771 
1 .841902 
2 842033 
3 842163 
4 842294 
5 842424 
6 842555 
V4 842685 
8 842815 
9 . 842946 
10 .843076 
11 | 9.848206 
12 . 843336 
13 843466 
14 . 848595 
15 843725 
16 .843855 
17 843984 
18 .844114 
19 .844243 
20 . 8443872 
21 | 9.844502 
22 .844631 
23 .844760 
24 .844889 
25 .845018 
26 845147 
97 845276 
28 .845405 
29 . 845533 
30 .845662 
831 | 9.845790 
82 .845919 
33 . 846047 
34 .846175 
35 . 846304 
36 . 846432 
37 .846560 
38 .846688 
39 .846816 
40 . 846944 
41 | 9.847071 
42 .847199 
43 . 847327 
44 .847454 
45 847582 
46 .847709 
47 . 847836 
48 .847964 
49 . 848091 
50 .848218 
51 | 9.848845 
52 848472 
53 . 848599 
54 848726 
55 848852 
56 | .848979 
57 . 849106 
58 . 849232 
5 849359 


134° Cosine. 








TABLE XXV.—LOGARITHMIC SINES 



































D. 1". || Cosine. | D.1". || ‘Tang. | D.1", | Cotang. 135° 
9.856934 9.984837 
we 5 10.015 
ae 856812 a "985090 59 hretsn 
se 856690} 3°53 985343 ee 014657 | 58 
514 ||. -856568 | S93 || -985596 | 4°55 | 014404 | 57 
ee 856446 |" 2-03 985848 | 4-20 014152 | 56 
anne “956323 | 2-08 986101 | 4°33 1013899 | 55 
ee “856201 11.5 9he "986354 ee "013646 | 54 
ie 856078 | $ p93 “986607 i "013393 | 53 
Sig “855956 | 5-03 "986860 a "013140 | 52 
Stig |p espess | Sire || raeia ese 012888 | 51 
oe ebUTLL. | So8 "987365 Pes "012635 | 50 
2.17. || 9-855588 | 9 ox || 9-987618 | 499 | 10.012882 |. 49 
wae 855465 | 30° "987871 hot "012129 | 48 
215 $5082 Eye 988123 pe ‘011877 | 47 
eG “855219 "98837 ‘ "011624 
Sere i Seaeogs fh Sane “988629 | 4:22 “011371 | 45 
San. |b 078 |) Bape -ogsss2 | 4-33 ‘011118 | 44 
bate 854850 | 3-12 989134 pies: "010866 | “43 
35 |t seer | Spe ggoss7 | 4-22 | [010613 | 42 
ae 854603) 5-8 "989640 | 4:28 "010360 | 41 
517 || -854480 | 3-92 || -980893 | 4-35 |  oto107 | 40 
9.854356 . || 9.990145 
hai 10. 009855 
vac Boles |. aa "990398 ie: ‘Conbee 4 
ae 854109 | 3-07 |) ‘900651 | 4-39 009349 | 37 
Be 853986 | 3-29 "990903 re 009097 | 36 
tag 853862 | 3-05 “991156 ye 008844 | 35 
a8 853738 | 3°54 "991409 Ss 008591 | 34 
ag 853614 | 3:07 991662 | 4-2 008338 | 33 
Bee 853490 | 3-50 991914 | 4° 0 “008086 | 32 
Sis || 933366) $:-0% |) ‘oozter | 4-22 | ‘007833 | Bt 
a4 s5se2 | 3-0! 992420 pe ‘007580 | 30 
9.853118 Bit i 
wae 9.992672 0.0 
aaa "852994 ra 992925 +8 4 rie oo 
a8 :e52g69 | 2-08 998178 | 4°33 “006822 | 27 
ae 852745 | 3-08 "993431 po, “006569 | 26 
573 || -852620 | §-92 || 998688 | 4:29 | 006817 | 25 
3°13. || -852496 | Sog || -993986 | fs, | 006064 | 24 
513 || 852871 | Sop || -994180 | 3°55 | 005811 | 23 
Sia. || 82247 | Som | 994441 | Srey |) 2008559 | oa 
a5 852122 Sing "994694 rs “005306 | 21 
aie i Wil) eae 994947 | 4°55 "005053 | 20 
tac 851872 9.995199 10.00480 
sue 851747 ane 995452 ie 004848 18 
o 43 851622 | $53 995705 vies "004295 | 17 ~ 
a 851497 | 2-08 995957 | 4-39 | -:004043 | 16 
aq 851372 | 3°45 "996210 Le "003790 | 15 
aad 851246 | 3-0) 996463 | 4° 2 003537 | 14 
as e5ltet | 308 996715 Tee "003285 | 13 
aie 7850996 | 5-95 "996968 age "003032 | 12 
ee -850870.| 3-1? “g97221 | 4:22 002779 | 11 
ae 850745 | £-08 997473 eg "002527 | 10 
9.850619 9.997726 
2.12 : , dé 10.002274 
Sie || 1850493 | 2:28 || ‘oover9 | 4-22 | ‘oogoet 3 
re 850368 | 5-45 998231 | 4°52 "001769 | 7 
Bd 850242 | 3-4 998484 | 4°55 ‘001516 | 6 
ae g50116 | 3-49 1998737 | 4°55 "001268 | 5 
iia pd 849990 | 5-15 998989 | 4°50 ‘0010611 | 4 
A "g49864 | 2: “999242 | 4-2 ‘000758 | 3 
ag | 2210 4.92 
snes 849611 | 5°15 “g99747 | 4: "000253 | 1 
9 849485 | *: 10.000000 | 4:7" | 40/000000 | 0” 
D. 1 Sine te Ge Cotang. | D. 1" Tang. |45° 





TABLE XXVI.—LOG. VERS. 8INES AND EX, SECS. 135 


i 


0° 








228481 


5.270859: 


.811266. 
349876 
.886843 
422300 
.456367 
489148 
520736 
551216 
580662 


5.609143 | 


636719 
663447 
. 689377 
. 714555 
. 739023 
762821 
785985 
808547 
830537 


5.851985 
872915 
893353 
913322 
. 932841 
951981 
970611 

5.988898 

6.006807 
024355 


6.041555 








q—2l 


oe 








173 
175 


EX. sec. 


2.626422 
\3., 228482 
- 580665 
(3, 830542 
4 024368 
182725 
316619 
432603 
634908 
626424 


4.709209 
(84787 
854312 
. 918681 

4.978608 

5.034666 
087825 
136972 
. 183935 
.228488 


5.270868 





5.852016 
872948 
893387 
913357 
. 932878 
951 on 

970652 

5.988940 

6.006851 
024401 


6.041602 
058470 





468180 
6.182780 


Inf. neg. 

















720C By 80 6.784741 




















1° 
te ‘| Vers. | q— 21 |Ex.sec. 
9.070 

3600 | 0 (6.182714 | 109 || 175 | 6.182780 
3660 | 1 197071 | 108 || 177| .197189 
3720 | 2 | .211194 | 108 || 179] .211264 
3780 | 8 | .225091 | 108 || 181] .225164 
3840 | 4 | .238770 | 107 || 182) .288845 
3900 | 5 | .252236 | 107 || 184] .252814 
3960 | 6 | .265497 | 106 || 186] .265577 
4020 | 7 | .278558 | 106)|188| .278641 
4080 | 8 | .291426 |106)\191| .291511 
4140 | 9 | .804106 |105/|/ 193] .304193 
4200 | 10 | .316603 | 105 || 195} .816693 
4260 | 11 |6.828923 | 104|| 197 | 6.329016 _ 
4820 | 12 | .841071 | 104||199| .841167 
4380 | 13 | .853052 | 103|| 201] .353150 
4440 | 14 | .864869 | 103 || 204| .36497 
4500 | 15 | .876528 | 103 || 206| .376631 
4560 | 16 | .888032 | 102)| 208} .888138 
4620 | 17 | .899386 | 102|| 211] .899494 
4680 | 18 | .410598 | 101 || 213] .410705 
4740 | 19 | .421657 | 101 || 215| .421772 
4800 | 20 | .482583 | 100 || 218| .482700 
4860 | 21 |6.443372 | 100 || 220 | 6.448493 
4920 | 22 | .454029 | 099 || 228) .454153 
4980 | 23 |. .464557 | 099 || 225 | .464684 
5040 | 24 | .474959 | 098 || 228) .475089 
5100 | 25 | .485238 | 098 || 230) .485371 
5160 | 26 | .495396 | 097 || 233 | .4955382 
5220 | 27 | .505488 | 097 || 286) .505577 
5280 | 28 | .515364 | 096 || 288] .515506 
5340 | 29 | .525178 | 095 || 241] .5253824 
5400 | 30 | .534882 | 095 |) 244} .535031 
5460 | 81 |6.544480 | 094 || 247 | 6.544632 
5520 | 82 | .553972 | 094|/ 249] .554128 
5580 | 83 | .563362 | 093 || 25%) .568521 
5640 | 84 | .572651 | 093 || 255] .572813 
5700 | 85 | .581842 | 092 || 258} .582008 
760 | 86 | .590936 | 092! 261] .591106 
5820 | 87 | .599987 | 091 |) 264] .600110 
5880 | 88 | .608845 | 090|) 267} .609021 
5940 | 89 | .617668 | 090|| 270} .617843 
6000 | 40 | .626392 | 089 || 273} .626575 
6060 | 41 |6.635034 | 089 || 276 | 6.635221 
6120 | 42 | .643591 | 088 || 279} .643782 
6180 | 43 | .652064 | 087 || 282! .652259 
6240 | 44 | .660456 | 087|| 285] .660655 
6300 | 45 | .668767 | 086 || 289| .668970 
6360 | 46 | .677000 | 085} 292} .677206 
6420 | 47 | .685155 | 085 || 295} .685365 
6480 | 48 | .693234 | 084|' 298] .693448 
6540 | 49 | .'701239 | 083) 3802] .701457 
6600 | 50 | .709171 | 083|| 805] .7093893 
6660 | 51 |6.717030 | 082 || 308 | 6.717257 
6720 | 52 | .724820 | 081 || 312} .725050 
6780 | 58 | .782540 | 081 || 315} .7382775 
6840 | 54 | .740192 | 080|| 819] .740431 
6900 | 55 | .747777 | 079 || 822] .748020 
6960 | 56 | .755297 | 079 || 826] 755544 
7020 | 57 | .762752 | 078|| 329] .763004 
7080 Be 770144 | 077 || 333} .'770400 
7140 . 777473 | 076 277733 
076 || 340 | 6.785005 





136 


TABLE XXVI.—LOGARITHMIC VERSED SINES 

























































































2° 8° 
CLE WEN NCXS, | g—2l | Ex. sec. Ad | | Vers. q—2l \Ex.sec. 
9,070 9.070* 
7200) 0|6.'784741 | 076]| 840/6.785005 |; 10800} 0)7.136868 } 021 616 |'7.137464 
7260| 1| .791948)| 075|| 344) .792217 || 10860] 1) .141679 019 622 | .142281 
7320} 2] .799096 | 074!) 848} .799870 10920} 2| .146464 018 627 | .147072 
7380! 3] .806186|073|| 351! .806464 10980} 3] .151222 017 633 | .151887 
7440) 4] .813219| 073}! 355| .813501 11040) 4] .155954 016 638 | .156577 
7500) 5| .820194|072)| 359] .820482 |} 11100} 5) .160661 015 644 | .161290 
7560) 6| .827115|071]|| 363) .827406 11160 6] .165342 014 650 | .165978 
620) 7| .833980|070]| 867| .834277 11220/ 7| .169998 013 655 | .170641 
%680| 8| .840792/070||371| .841093 11280} 8] .174630 011 661 | .175279 
.'7740| 9] .847551 | 069]| 375| .847857 || 11840] 9) .179236 010 667 | .179893 
%800|10| .854257 | 068|| 379} .854568 || 11400/10} .183819 | 009 673 | .184483 
7860 11 6.860912 | 067 || 383 6.861228 || 11460/11/7.188377 | 008 679 |'7%.189048 
7920|12| .867517 | 066!| 887] .867837 11520/12] .192912 007 685 | .193589 
7980/13] .874071]066|| 391| .874396 11580 |18] .197423 006 690 | .198108 
8040 | 14} .88057'7 | 065!| 395} .880907 11640}14} .201910 004 696 | .202602 
8100/15} .887034 | 064!| 399] .887369 11700 |15}| .206375 003 702 | .207074 
8160/16} .893443 | 063]| 403) .893783 11760 |16} .210817 002 708 | .211523 
8220/17} .899806 | 062|| 407} .900151 |} 11820)17] .215236 | 001 714 | .215949 
8280) 18} .906122|061|| 411) .906472 |} 11880}18} 219633 *00 720 | .220853 
8340/19! .912393|061}| 416} .912748 11940 /19} .224007 998 (279) 224785 
8400/20) .918618] 060/| 420) .918979 || 12000}20| .228360 | 997 733 | .229095 
8460 | 21 |6.924800 | 059|| 42416.925165 || 12060 | 21 |'7.282691 996 739 |'7 2338433 
8520) 22) .9380937 | 058|| 429} .931808 12120 |22| .237000 995 745 | .237750 
8580 |23| .937032 | 057|| 433) .937408 12180 | 23} .241288 994 751 | 242046 
8640) 24) .9438084! 056!| 437) .943465 12240 | 24]  ..245555 992 "57 | , 246320 
8700} 25} .949094|055|| 442) .949480 12800 | 25) .249801 g91 "64 | .2505'74 
8760/26} .955063 | 0541| 446] .955455 12360 |26| .254027 990 770 | .254807 
8820) 27| .960991 | 054!| 451} .961388 12420 |27) .258282 989 776 | .259019 
8880) 28] .966879|053]| 455] .967281 12480 }28] .262416 987 783 | .263212 
8940/29) .972727]| 0521| 460| .973135 12540 }29} .266581 986 789 | .2673884 
9000) 80) .978536) 051]! 464] .978949 || 12600/80| .270726 | 985 795 | .271537 
9060 | 31 |6.984306 | 050!| 469|6.984725 || 12660|31|7.274851 | 983 802 |7. 275669 
9120 32} .990039 | 049|| 474] .990463 12720 |32| .278956 982 808 | .2797838 
9180) 33 |6.995738 | 048/| 478 /6. 996164 12780 | 33] .283043 981 815 | .288877 
9240} 34|7.001391 | 047|| 483)7.001827 || 12840/34] .287110 | 980 821 | .287952 
9300/35] .007012] 046]| 488) .007454 |! 12900/35| .291158 | 978 828 | .292007 
9360) 36| .012597] 045|| 493] .013044 12960 |36] .295187 977 8385 | .296045 
9420) 37} .018146| 044|| 497) .018599 18020 |37| .299197 976 841 | .800063 
9480| 38} .023660 | 043}| 502) .024119 13080 |38] .803190 974 848 | .804063 
9540| 39} .029139| 042]| 507} .029604 13140|39| .807164 973 855 | .3808045 
9600) 40} .034584| 041)! 512} .085054 18200 |40| .811119 972 861 | .812009 
9660) 41 |'7.039995 | 040)! 51'7|7 .040471 || 18260)41|7.315057 | 970 868 |'7.3815955 
9720142} .045372/ 039|| 522) .045854 18320 |42| .318977 969 87 .3819883 
9780) 43} .050716} 038|| 527) .051204. 13380 | 43] .822880 967 882 | .323794 
9840) 44} .056028 | 037|| 532) .056522 18440 |44| .826765 966 889 | .3827687 
9900| 45} .061307| 036|| 537] .061807 13500 |}45| .830632 965 896 | .331563 
9960/46} .066554| 035|| 542) .067061 13560|46| .334483 963 902 | .3385422 
10020| 47} .071770)| 084}| 547} .072282 18620 |47| .3383816 962 908 | .3389263 
10080} 48} .076954| 033)| 552) .077473 18680 |48| .342133 961 916 | 343089 
10140} 49} .082108) 0321) 557} .082633 138740 }49| .345933 959 928 | .346897 
10200} 50) .087232)| 031|| 562) .0877638 18800 |50|] .3849716 958 930 | .3850689 
10260) 51 |'7.092325 | 030)| 568 7.092862 || 13860|51/7.853483 | 956 988. |7 354464 
10320) 52] .097389) 029|| 573} .097932 18920 |52| .357233 955 945 | .358223 
10380) 53} .102423) 028||578) .102973 13980 /53] .360968 953 952 | .3861966 
10440| 54} .107428) 027/)| 584} .107985 14040 |54| .364686 952 959 | .365693 
10500) 55] .112405) 026)| 589} .112968 14100 |55|] .368389 951 966 | .869404 
10560 | 56} .117353) 025||594} .117922 || 14160/56] .372076 949 973 | .3873100 
10620/57} .122273| 024|| 600} .122849 14220/57| .3875747 948 981 | .3876780 
10680| 58} .127165| 023|| 605) .127748 || 14280158) .379403 946 988 | .380444 
10740|59} .182030] 022} 611) .132619 14340/59} .3838043 945 995 | .3884094 
10800 | 60) 7.186868 | 021} | 616|'7.137464 || 14400/60|7.386668 | 943 *03 (7.387728 
ees | rath 9.069*| 9.071* 


. 


AND EXTERNAL SECANTS 


137 








/ Vers. 
0 | 7.886668 
1 .890278 
2 .398874 
3 397454 
4 .401020 
5 .404571 
6 .408107 
7 .411629 
8 .415137 
9 .418631 
10 .422111 
deh 7 42bo 0G 
12 .429029 
13 .432467 
14 .435892 
15 .439303 
16 .442701 
17 .446086 
18 .449458 
19 .452816 
20 .456162 
21 | 7.459495 
ze .462815 - 
23 .466122 
24 .469417 
2D .472699 
26 .475969 
27 479226 
28 .482472 
29 .485705 
30 -488927 
31 | 7.492136 
32 .495333 
33 -498519 
o4 .501693 
35 .604856 
36 .508007 
37 .511147 
38 .514275 
39 .517392 
40 » 520498 
41 | 7.523593 
42 7526677 
43 529750 
44 532812 
45 .535863 
46 . 538904 
{6 .5419384 
48 .544953 
49 .547962 
50 .550961 
51 | 7.553949 
52 556927 
53 .559895 
54 . 562852 
55 . 565800 
56 . 568737 
57 .571665 
58 .574583 
59 577491 
_ 60 | 7.580889 














Ex, sec. 


7.387728 
391347 
.394951 
898540 
402114 
405674 
409220 
412751 
416268 
419771 
423260 


7.426735 


"454050 


-~ 
iS 
S 
mS 
ie 2) 


490267 


7.493487 
496694 
.499890 
508074 
506247 
. 509408 
512558 
915697 
518824 
521940 


525046 
.528140 
531228 
534296 
537357 
.540408 
543449 
546479 
549499 
552508 
555507 


per 


~ 


7.582045 | 


Dale 


60.32 
60.07 
59.82 
59.57 














Lome ~ 
COMIRHUIP WMHS 








Vers. 


7.580389 
583278 
586157 
-589026 
591886 
594737 
597578 
.600410 
6032383 
606047 
608851 


.611647 
614433 
617211 
.619980 
622739 
625491 
628233 
630967 
683692 
-636409 


7.639117 
641816 
644508 
-647191 
649865 
652532 
.655190 
657840 
-660483 
663117 


7.665743 
668361 
670971 
673574 
.676168 
678755 
681334 
.683906 
-686470 
-689026 


.691575 
-694116 
. 696650 
699177 
701696 
704208 
(06712 
.709210 
711700 
714183 
7.716659 
719128 
.721590 
- 724045 
. 726493 
«728934 
731368 
.733796 
. 736216 
7.738630 


I 


=t 








5° 
D, 1”. | Ex. sec. 
48.15 |'7.582045 
47.98 | .584945 
47.82 | .587835 
47.67 | .590715 
47.52 | .593587 
47.35 | .596449 
47.20. | .599301 
47.05 | .602144 
46.90 | .604979 
46.73 | .607804 
46.60 | .610619 
46.48 |7.613426 
46.30 | .616224 
46.15 | .619013 
45.98 | .621794 
45.87 |. .624565 
45.70 | .627328 
45.57 | .6380082 
45.42 | .682827 
45.28 | .685564 
45.138 | .688293 
44.98 |7.641013 
44.87 | .6438724 
44.72 | .646428 
44.57 | .649122 
44.45 | .651809 
44.30 | .654488 
44.17 | .657158 
44.05 | .659820 
43.90 | .662474 
43.77 | .665121 
43.63 |7.667759 
43.50 | .670389 
43.38 | .673012 
43.23 | .675626 
43.12 | .678233 
42.98 | .680833 
42.87 | .683424 
42.73 | .686008 
42.60 | .688585 
42.48 | .691154 
42.35 |7.698715 
42.2 .696269 | 
42.12 | .698815 
41.98 | .701355 
41.87 | .703887 
41.7 706411 
41.63 | .708929 
41.50 | .711439 
41.38 | .7138942 
41.27 | .716488 
41.15 |7.718927 
41.08 | .721409 
40.92 | .723884 
40.80 | .726852 
40.68 | .728813 
40.57 | .731267 
40.47 | .738714 
40.83 | .736155 
40.23 | .738589 
40.13 |7.741016 








a 


138 


TABLE XXVI.—LOGARITHMIC VERSED SINES 


J 





COWIDOTA OWL O 


10 








790102 


7.810308 
812524 
814734 
816939 
819139 
821332 
823521 
825708 
827880 
830052 


7.832218 
834379 
836535 
838685 
840830 
842969 
845104 
847233 
849357 
851475 


7.853589 


PHSECE, 








6° 


D, 1". | Ex. sec. 








40.18 | 7.741016 
40.00 743436 
39.90 | .745850 
89.78 | .748258 
89.68 |  .750658 
39.57 753052 
39.47 755440 
39.35 757821 
39.25 760196 
39.18 762565 
39.05 764927 
38.92 | 7.767282 


769632 
771975 
774312 
776643 

- 778968 
781286 
783599 
785905 
788206 


%.835180 
837356 
839526 
841691 
843851 
846005 
848155 
850299 
852437 
854571 


7.856700 
858823 
860942 
863055 
865163 
867266 
869365 

.871458 
873546 
84.38 | 7.875630 





D. 1", 


40.33 
40.23 
40.13 
40.00 
39.90 
39.80 




















930297 
932227 


%.934152 
936073 
937990 
939903 
941811 
943715 
945615 
947511 
949402 
951290 


%.953178 
955052 
956928 
958799 
960666 
962529 
964388 
966248 
968094 
969941 


7.971785 
973624 
975459 








934012 
935958 


7.937900 
939838 





972148 
974013 


7. 975874 
730 
9583 
981432 
983277 
985119 
986956 
988790 
30.13 | .9906: 
30.08 '7..992446 











We 
Vers. | D. 1". | Ex. sec./D. 1”. 
7.872381 | 34.38 |7.875630 | 34:63 
874444 | 34.30 | .877708 | 34.57 
876502 | 34.22 | .879782 | 34.48 
878555 | 34.13 | .881851 | 34.40 
880603 | 34.07 | .883915 | 34.32 
882647 | 33.98 | .885974 | 34.25 
884686 | 33.90 888029 | 34.15 
.886720 | 83.82 | .89007' 34.08 
.888749 | 33.73 | .892123 | 34.02 
.890773 | 83.67 | .894164 | 33.92 
.892793 | 33.58 | .896199 | 33.85 
fh 894808 83.50 |'7.898230 | 33.77 


AND EXTERNAL SECANTS 139 


rr 


8° 9° 





4 | Vers. | D.1°.| Ex. sec. | D. 1’. Vers. | D.1’.| Ex. sec./D. 1”. 


—  |—___. 





0 | 7.988199 | 30.08 | 7.992446 | 30.38 
1 990004 | 30.02 | .994269 | 30.32 
2} .991805 | 29.95 .996088 | 30.25 
3} .993602 | 29.88 | .997903 | 30.18 
: 995395 | 29.83 | 7.999714 | 30.13 
6 
v 
8 
9 








8.090317 | 26.72 |8.095697 | 27.05 
091920 | 26.68 | .097820 | 27.02 
.093521 | 26.63 | .098941 | 26.97 
.095119 | 26.58 | .100559 | 26.92 
.096714 | 26.52 | .102174 | 26.87 

ie : 26.82 
.099894 | 26.43 | .105395 | 26.77 
-101480 | 26.40 | .107001 | 26.73 


.997185 | 29.77 | 8.001522 | 30.07 
7.998971 | 29.72 | .008326 | 30.00 
8.000754 | 29.63 | .005126 | 29.95 





—_ 

SOMMIRHMIRWWHOS | pS 
fom) 
iJon} 
(ea) 
co 
[om] 
or 
PS) 
[on] 
— 
o 
EO) 
~2 
@ 
[or 


10 006079 29.47 “010506 29.73 .106221 | 26.25 | .111803 | 26.58 


11 | 8.007847 | 29.40 | 8.012292 | 29.70 || 11 | 8.107796 | 26.18 [8.113398 | 26.53 
12 | .009611 | 29.35 -014074 | 29.65 || 12 | .109367 | 26.15 | .114990 | 26.48 
13 | .011872 | 29.28 -015853 | 29.58 || 13 | .110936 | 26.10 | .116579 | 26.45 
14 | .013129 | 29.22 017628 | 29.53 || 14 112502 | 26.05 | .118166 | 26.38 
15 | .014882 | 29.17 -019400 | 29.47 || 15 | .114065 | 26.00 | .119749 | 26.35 
16 | .016632 | 29.10 -021168 | 29.42 || 16 | .115625 | 25.95 | .121380 | 26.30 
17 | .018878 | 29.05 022933 | 29.85 || 17 | .117182 | 25.92 | .122908 | 26.25 
18 | .020121 | 29.00} .024694 | 29.30 || 18 | .118787 | 25.87 | .124483 | 26.22 
19 | .021861 | 28.93 -026452 | 29.23 || 19 120289 | 25.82 | .126056 | 26.17 
20 | .028597 | 28.87 028206 | 29.18 || 20 | .121888 | 25.77 | .127626 | 26.12 


21 | 8.025829 | 28.82 | 8.029957 | 29.13 || 21 | 8.128384 | 25.72 |8.129193 | 26.07 
22 | .027058 | 28.75 -031705 | 29.07 || 22 | .124927 | 25.68 | .180757 | 26.02 
23 | 0287 28.70 | .033449 | 29.00 || 23 126468 | 25.65 | .182318 | 25.98 
24 | .080505 | 28.65 | .035189 | 28.97 || 24 | .128006 | 25.58 | .133877 | 25.93 
25 | .082224 | 28.58 | .036927 | 28.90 || 25 129541 | 25.55 | .185433 | 25.90 
26 | .033939 | 28.53 | .038661 | 28.83 || 26 | .181074 | 25.50 | .186987 | 25.85 
27 | .085651 | 28.47 | .040391 | 28.78 || 27 | .182604 | 25.45 | .1388538 | 25.80 
28 | .087359 | 28.42 | .042118 | 28.7 28 .184131 | 25.40 | .140086 | 25.75 
29 | .039064 | 28.37 | .043842 | 28.68 || 29 | .135655 | 25.387 | .141631 | 25.72 
30 | .040766 | 28.30 | .045563 | 28.62 || 30 | .187177 | 25.82 | .143174 | 25.67 


31 | 8.042464 | 28.25 | 8.047280 | 28.57 || 31 | 8.188696 | 25.27 |8.144714 | 25.63 
32 | .044159 | 28.20 | .048994 | 28.50 || 82] .140212 | 25.23 | .146252 | 25.58 
33 | .045851 | 28.13 | .0507 28.47 || 83 | 141726 | 25.18 | .147787 | 25.58 
34 | .047539 | 28.08 | .052412 | 28.40 || 84] .148287 | 25.13 | . 149319 25.50 
35 | .049224 | 28.03 | .054116 | 28.35 || 85 | .144745 | 25.10 | .150849 | 25.45 
36 | .050906 | 27.98 | .055817 | 28.28 || 36 | .146251 | 25.05 | .152376 | 25.40 
BY | 052585 | 27.92) .057514 | 28.25 || 87] .147754 | 25.02 | .153900 | 25.37 
38 | .054260 | 27.87 | .059209 | 28.18 || 88 | .149255 | 24.95 | .155422 | 25.33 
39 | .055932 | 27.82 | .060900 | 28.13 || 89 | .150752 | 24.93 | .156942 | 25.27 
40 | .057601 | 27.75 | .062588 | 28.08 || 40 | .152248 | 24.88 | .158458 | 25.25 


41 | 8.059266 | 27.72 | 8.064273 | 28.03 || 41 | 8.153741 | 24.83 |8.159973 | 25.48 
42 | .060929 | 27.65 065955 | 27.97 || 42 | .155231 | 24.78 | .161484 | 25.17 
43 | .062588 | 27.60 | .067633 | 27.93 || 43 | .156718 | 24.75 | .162994 | 25.10 
44 | .064244 | 27.551 .069309 | 27.87 || 44 | .158203 | 24.7 -164500 | 25.07 
45 | .065897 | 27.48 | .070981 | 27.82 || 45 | .159686 | 24.67 | .166004 | 25.08 
46 | .067546 | 27.45 072650 | 27.77 || 46 | .161166 | 24.62 | .167506 | 24.98 
47 | .069193 | 27.38 } .074316 | 27.72 || 47 | 162643 | 24.58 | .169005 | 24.95 
48 | .070836 | 27.33 | .075979 | 27.67 || 48 | .164118 | 24.53 | .170502 | 24.90 
49 | .072476 | 27.30 | .07'7639 | 27.60 || 49 | .165590 | 24.50 | .171996 | 24.87 
50 | .Or4114 | 27.28 | .079295 | 27.57 || 50 | + .167060 | 24.45 | .173488 | 24.82 


51 | 8.075748 | 27.18 | 8.080949 | 27.52 || 51 | 8.168527 | 24.42 [8.174977 | 24.78 
52 | .077379 | 27.13 | .082600 | 27.45 || 52 | .169992 | 24.387 | .176464 | 24.73 
53 |  .079007 | 27.07%» .084247 | 27.42 || 53 | .171454 | 24.83 | .177948 | 24.7 

54] .080631 | 27.03 | .085892 | 27.37 || 54 | .172914 | 24.80 | .179430 | 24.65 
55 | .082253 | 26.98} .087 27.30 || 55 | 1743872 | 24.25 | .180909 | 24.62 
56 | .083872 | 26.93 | .089172 | 27.27 || 56 | .175827 | 24.20 | .182386 | 24.58 
57 | 085488 | 26.87 | .090808 | 27.20 || 57 | 177279 | 24.17 | .183861 | 24.53 
58} .087100 | 26.83 | .092440 | 27.17 || 58 | 178729 | 24.18 | .185333 | 24.50 
59 | .088710 | 26.78 | .094070 | 27.12 || 59 | .180177 | 24.08 | .186803 | 24.47 
60 | 8.090317 | 26.72 | 8.095697 1 27.05 |! 60 | 8.181622 | 24.05 18. 188271 | 24.42 
al 













































































140 ‘TABLE XXVI.—LOGARITHMIC VERSED SINES 
10° gL 

Vers. | D.1". | Ex. sec. | D. 1°". |] Vers. D. 1". | Ex, see.|D. 1%. 
O | 8.181622 | 24.05 | 8.188271 | 24.42 || 0 | 8.264176 | 21.85 |8.279929 | 22.97 
1 | .183065 | 24.00] .189786.| 24.37 || 1] .265487 | 21.82 | .273565 | 22.22 
2] .184505 | 23.97 | .191198 | 24.35 || 2] .266796 | 21.78 | .274898 | 22.20 
- 3] .185943 | 23.93 | .192659 | 24.30 || 3 268103 | 21.75 | .276230 | 22.17 
4] .187379 | 23.88 | .194117 | 24.25 || 4 269408 | 21.72 77560 | 22.13 
5 | .188812 | 23.85 | .195572 | 24.22 || 5 270711 | 21.68 | .278888 | 22.08 
6 | .190243 | 23.80] .197025 | 24.18 6 272012 | 21.65 | .280213 | 22.07 
"| .191671 | 23.7 198476 | 24.15 || 7 273311 | 21.62 | .281537 | 22.03 
8 | .193097 | 23.73 | .199925 | 24.10 || 8 274608 | 21.58 | .282859 | 22.00 
9| .194521 | 23.68 | .201371 | 24.07 || 9 tes 21.57 | .284179 | 21.98 
10} .195942 | 23.65} .202815 | 24.08 || 10 277197 | 21.52 | 285498 | 21.93 
11 | 8.197361 | 23.62 | 8.204257 | 23.98 || 11 | 8.278488 | 21.48 |8.286814 | 21.90 
12] .1987'78 | 23.57 | .205696 | 23.95 || 12] .279777 | 21.47 | .288198 | 21.88 
13 | .200192 | 23.53 | .207133 | 23.92 || 13 281065 | 21.42 | .289441 | 21.83 
14} 201604 | 23.50 | .208568 | 23.88 || 14 282350 | 21.40 | .290751 | 21.82 
15 |} .208014 | 23.45 | .210001 | 23.83 || 15 283634 | 21.37 | .292060 | 21.7 
16 | .204421 | 23.42 | .211431 | 23.80 || 16 284916 | 21.33 | .293367 | 21.75 
@ | .205826 | 23.38 | .212859 | 23.77 || 17 286196 | 21.28 | .294672 | 21.72 
18 | .207229 | 23.35 | .214285 | 23.72 || 18 | .2987473 | 21.297 | .295975 | 21.68 
19 | .208630 | 23.30 | .215708 | 23.70 || 19 | .288749 | 21.95 | .2907976 | 21.67 
20] .210028 | 23.27 217130 | 23.65 || 20 | .290024 | 21.20 | .298576 | 21.62 
21 | 8.211424 | 23.23 | 8.218549 | 23.62 || 24 | 8.291996 | 21.17 |8.299873 | 21.60 
22] .212818 | 23.18} .219966 | 23.57 || 22 | .292566 | 21.15 | .301169 | 21.57 
23 | .214209 | 23.17 |  .221380 | 23.55 || 23 | 293835 | 21.10 | .302468 | 21.53 
24{ .215599 | 23.12 | .222793 | 23.50 || 24 | .295101 | 21.08 | .308755 | 21.50 
25 | .216986 | 23.08 | .224203 | 23.47 || 25 | .296366 | 21.05 | .305045 | 21.48 
26 | .218371 | 23.03 | .225611 | 23.43 || 26 | .297629 | 21.02 | .306334 | 21.43 
27 | .219753 | 23.00 | .227017 | 23.40 || 27 | 298890 | 20.98 | .307620 | 21.42 
283 | 221183 | 22.98 | .228421 | 23.35 || 28 | 300149 | 20.95 | .308905 | 21.38 
29 | .222512 | 22.93 | .2290822 | 23.32 || 29 | .801406 | 20.93 | .310188 | 21.35 
30 | .223888 | 22.88 | .231221 | 23.30 || 30 | .302662 | 20.90 | .311469 | 21.33 
31 | 8.225261 | 22.87 | 8.232619 | 23.25 || 31 | 8.303916 | 20.85 18.312749 | 21.28 
32 | .226633 | 22.82 | .234014 | 23.22 || 82 |} .305167 | 20.85 | .314026 | 21.97 
33 | 228002 | 22.78 | .235407 | 23.17 || 83] .306418 | 20.80 | .315302 | 21.23 
34 | .229369 | 23.77 | .236797 | 23.15 || 34 | .807666 | 20.77 | .316576 | 21.22 
35 | .230735 | 22.70 | .238186 | 23.10 || 85 | 308012 | 20.75 | 1317849 | 21 17 
36 | ,232097 | 22.68 | .239572 | 23.08 || 36 | .310157 | 290.72 | 1319119 | 21.15 
37 | 233458 | 22.65 | .240957 | 23.03 || 37 | .311400 | 20.68 | 320888 | 21.12 
38 | 234817 | 22.60 | _.242339 | 23.00 || 88 | 319641 | 20.65 |. .391655 | 21.08 
39 | 236173 | 22.57 | .243719 | 22.97 || 89 | .313880 | 20.62 | .322990 | 21.05 
40 | 237527 | 22.55 | .245097 | 22.93 || 40 | .815117 | 20.60 | .324183 | 21.03 
41 | 8.238880 | 22.50 | 8.246473 | 22.90 || 41 | 8.816853 | 20.57 |8.325445 | 21.00 
42 | .240230 | 22.47 | 247847 | 22.87 || 42 | 317587 | 20.53 |. 326705 | 20.98 
43 | .241578 | 22.43 | .249219 | 22.83 || 43 | .318819 | 20.50 | /327964 | 20.93 
44) 1242924 | 22.388 | .250589 | 22.80 || 44 | [320049 | 20.48 | 1329220 | 20.92 
45 | 244967 | 22.37 | .251957 | 22.75 || 45 | 1391278 | 20.45 | 1330475 | 20.88 
46 | .245609 | 22.32 | .253322 | 22.73 || 46 | 329505 | 20.42 | 1331728 | 20.87 
47 | 246948 | 22.30 | 254686 | 22.68 || 47 | .323730 | 20.38 | 332980 | 20.82 
48 | 1248286 | 22.25 | .256047 | 22.67 || 48 | .324953 | 20.37 | 1334229 | 20.80 
49 | 249621 | 22.93 | .257407 | 22.62 || 49 | .396175 | 20.33 | 1335477 | 20.78 
50] .250955 | 22.18 | .258764 | 22.60 || 50] .327395 | 20.30 | 336724 | 20.73 
51 | 8.252286 | 22.15 | 8.260120 | 22.55 |! 51 | 8.328613 | 20.27 |8.3837968 | 20.79 
52 | .258615 | 22.12 | .261473 | 22.53 || 52) .329899 | 20.25 | .339211 | 20.70 
53 | .254942 | 22.10 | .262825 | 22.48 || 53 331044 | 20.22 | .3840453 | 20.65 
54] 1256268 | 22.05 | .264174 | 22.47 || 54 832257 | 20.18 | .341692 | 20.63 
55 | .257591 | 22.02 | 265592 | 22.42 || 55 333468 | 20.17 | .342930 | 20.60 
56 | 258912 | 21.98 | .266867 | 22.40 || 56 834678 | 20.13 4 .344166 | 20.58 
57 | = .260231 | 21.95 | .268211 | 22.35 || 57 835886 | 20.10 | 345401 | 20.55 
58} .261548 | 21.92 | .269552 | 22.33 || 58 337092 | 20.07 | .346634 | 20.52 
59 | 1262863 | 21.88 | .270892 | 22.28 || 59] 3388296 | 20.05 | .347865 | 20.50. 
GO | 8.264176 | 21.85 | 8.272229 | 22.27 Il 60 | 8.339499 | 20.02 18.349095 | 20.47 


: ; 


AND EXTERNAL SECANTS 141 
ce rt NE RE 



























































12° 13° 
ei Vers. Dil’) Exesec. | D1". |)'*7'| ° Vers: | Dil .\f Ex} siellD. 1’, 
0 | 8.339499 | 20.02 | 8.349095 | 20.47 0 | 8.408748 | 18.47 |8.420024 | 18.95 
1 .340700 | 20.00 .350323 | 20.43 1 -409856 | 18.43 | .421161 | 18.93 
2 .3841900 | 19.95 .801549 | 20.42 2 .410962 | 18.42 | .422297 | 18.90 
3 .845097 | 19.95 802774 | 20.38 3 -412067 | 18.40 | .423431 | 18.88 
4 344294 |1 49.90 .393997 | 20.35 4 413171 | 18.88 | .424564 | 18.87- 
5 .845488 | 19.88 .855218 | 20.33 5 -414274 | 18.35 | .425696 | 18.83 
6 3846681 | 19.85 .306438 | 20.30 6 .415375 | 18.382 | .426826 ; 18.82 
7 847872 | 19.82 .857656 | 20.28 ff .416474 | 18.30 | .427955 | 18.80 
8 .349061 | 19.80 | .3858873 | 20.25 8 417572 | 18.28 | .429083 | 18.77 
9 3850249 | 19.77 | .860088 | 20.22 9 -418669 | 18.25 | .430209 | 18.75 
10 | .351485 | 19.75 | .861801 | 20.20 || 10 | .419764 | 18.23 | .431334 | 18.73 
11 | 8.352620 | 19.72 | 8.362513 | 20.18 || 11 | 8.420858 | 18.22 |8.432458 | 18.70 
12 .393803 | 19.68 .863724 | 20.13 || 12 -421951 | 18.18 | .483580 | 18.67 
13 .354984 | 19.67 .064982 | 20.12 || 13 423042 | 18.17 | .434700 | 18.67 
14 .356164 | 19.63 .366139 | 20.10 || 14 -424132 | 18.13 | .4385820 | 18.63 
15 .807842 | 19.60 .867345 | 20.07 || 15 .425220 | 18.12 | .486988 | 18.62 
16 .3858518 | 19.58 .868549 | 20.03 || 16 .426307 | 18.10 | .488055 | 18.58 
17 | 359693 | 19.55 .869751 | 20.02 || 17 -427393 | 18.07 | .489170 | 18.57 
18 .3860866 | 19.58 .870952 | 19.98 || 18 .428477 | 18.05 | .440284 | 18.55 
19 .862038 | 19.50 .872151 | 19.95 || 19 -429560 | 18.02 | .441397 | 18.53 
20 .863208 | 19.48 .3873348 | £9.95 || 20 -430641 | 18.02 | .442509 | 18.50 
21 | 8.364877 | 19.43 | 8.374545 | 19.90 || 21 | 8.431722 | 17.97 |8.443619 | 18.47 
22} .365542 | 19.43 | -.875789 | 19.88 || 22 -432800 | 17.97 | .444727 | 18.47 
23 .366709 | 19.38 .876932 | 19.85 || 23 .433878 | 17.93 | .445885 | 18.43 
24 .867872 | 19.387 |. .878123 | 19.88 || 24 .434954 | 17.92 | .446941 | 18.42 
25 369034 | 19.85 .879318 | 19.82 || 25 .436029 | 17.88 | .448046 | 18.38 
26 70195 | 19.82 .880502 | 19.78 || 26 .437102 | 17.87 | .449149 | 18.38 
27 871354 | 19.28 .881689 | 19.75 || 27 -438174 | 17.85 | .450252 | 18.35 
28 .8/2511 | 19.27 .3882874 | 19.73 || 28 .439245 | 17.82 | .451853 | 18.32 
29 .3873667 | 19.25 .384058 | 19.70 |) 29 .440314 | 17:80 | .452452 | 18.32 
30 | .3874822 | 19.20 | .3885240 | 19.68 || 80 | 4413882 | 17.78 | .453551 | 18.28 
81 | 8.375974 | 19.18 ; 8.386421 | 19.65 || 31 | 8.442449 | 17.75 8.454648 | 18.25 
382 .307125 | 19.17 .387600 | 19.63 || 32 443514 | 17.7 .455748 | 18.25 
33 878275 | 19.13 | .3888778 | 19.60 || 33 .444578 | 17.72 | .4568388 | 18.22 
34 .379423 | 19.12 389954 | 19.58 || 34 .445641 | 17.68 | .457931 | 18.20 
35 .3880570 | 19.08 ~891129 | 19.55 || 385 .446702 | 17.68 | .459023 | 18.18 
36 .881715 | 19.05 .892302 | 19.53 || 86 .447763 | 17.63 | .460114 | 18.15 
37 882858 | 19.03 893474 | 19.50 || 37 -448821 | 17.63 | .461203 | 18.18 
38 .384000 | 19.02 394644 | 19.48 || 388 -449879 | 17.62 | .462291 | 18.12 
39 .385141 | 18.98 -895813 | 19.45 || 39 .450935 | 17.58 | .463378 | 18.10 
40 .886280 | 18.95 -896980 | 19.43 || 40 .451990 | 17.55 | .464464 | 18.07 
41 | 8.387417 | 18.93 | 8.398146 | 19.42 || 41 | 8.453043 | 17.55 |8.465548 | 18.05 
42 2388553 | 18.92 -899311 | 19.38 || 42 .454096 | 17.52] . ee 18.03 
43 .389688 | 18.88 .400474 | 19.85 || 43 .455147 | 17.48 | 4677 18.00 
44 .390821 | 18.85 .401635 | 19.33 || 44 .456196 | 17.48 “468703 18.00 
45 .891952 | 18.83 402795 | 19.82 || 45 .457245 | 17.45 | .469873 | 17.97 
46 -393082 | 18.82 .403954 | 19.28 || 46 .458292 | 17.43 | .470951 | 17.95 
47 .394211 | 18.48 -405111 | 19.27 || 47 .4593388 | 17.40 | .472028 | 17.92 
48 3895338 | 18.75 .406267 | 19.23 || 48 .460382 | 17.40 | .473103 | 17.90 
49 .396463 | 18.7 .407421 | 19.22 |) 49 .461426 | 17.37 | .474177 | 17.90 
50 | = .897587 | 18.72 408574 | 19.18 || 50 462468 | 17.35 | .475251 | 17.85 
51 | 8.398710 | 18.68 | 8.409725 | 19.17 || 51 | 8.463509 | 17.32 |8.476322 | 17.85 
52 .3899831 | 18.67 .410875 | 19.18 || 52 -464548 | 17.30 | 477393 | 17.83 
53 .400951 | 18.63 | .412023 | 19.18 || 53 .465586 | 17.28 | .478463 | 17.80 
54 .402069 | 18.62 .413171 | 19.08 || 54 -466623 | 17.27 | .479531 | 17.78 
55 -403186 | 18.58 .414316 | 19.08 || 55 .467659 | 17.23 | .480598 | 17.77 
56 .404301 | 18.57 -415461 | 19.03 || 56 .468693 | 17.23 | .481664 | 17.78 
- 57 -405415 | 18.53 416603 | 19.03 || 57 469727 | 17.20 | .482728 | 17.73 
58 .406527 | 18.52 .417745 | 19.00 || 58 pa hae 17.17 | .483792 | 17.70 
59 .407638 | 18.50 | .418885 | 18.98 || 59 .471789 | 17.17 | .484854 | 17.68 
60 | 8.408748 | 18.47 | 8.420024 | 18.95 || 60! 8. 419819 17.13 |8.485915 | 17.67 





142 


TABLE XXVI. LOGARITHMIC VERSED SINES 





4 Vers. 
0 | 8.472819 
1 .473847 
2 474874 
3) .475900 
4 . 476925 
5 477948 
6 .478970 
{4 .479991 
8 .481011 
9 .482029 
10 .485046 
11 | 8.484062 
12 -485077 
13 .486091 
14 .487103 
15 .488115 
16 .489125 
17 .490134 
18 .491141 
19 .492148 
20 .498153 
21 | 8.494157 
22 .495160 
PBI .496162 
24 .497162 
25 .498162 
26 .499160 
27 . 500157 
28 .501153 
29 .502148 
380 .503142 
31 | 8.504134 
32 .505125 
33 .506116 
34 .507105 
35 .508092 
36 .509079 
37 .510065 
388 .511049 
39 .5120388 
40 . 518015 
41 | 8.513996 
42 514976 
43 ,515955 
44 .516932 
45 .517909 
46 .518884 
47 .519859 
48 . 520882 
49 .521804 
50 522075 
51 | 8.523745 
52 524714 
53 . 525682 
54 . 526648 
55 .527614 
56 .528578 
Be 529542 
58 580504. 
59 .531465 
60 | 8.532425 


‘14° 


Die 


17.18 
17.12 
17.10 
17.08 
17.05 
17.03 
17.02 
17.00 
16.97 
16.95 
16.93 


16.92 
16.90 
16.87 
16.87 
16.83 
16.82 
16.78 
16.78 
16.75 
16.73 


16.72 
16.70 
16.67 
16.67 
16.63 
16.62 
16.60 


Ex, gece. 





8.485915 


486975 
.488033 
-489091 
490147 
:491202 
492256 
:493308 
-494360 
495410 
496459 


8.497507 
498554 
499600 
.500644 
.501687 
502730 
50877 
.504810 
505849 
506887 


8.507923 
508958 
509993 
.511026 
.512057 
.518088 
-514118 
-515146 
516174 

517200 


8.518225 
.519249 
-520272 
521294 
522815 
523334 
524853 
.525870 
- 526387 
527402 


8.528416 
529429 
.530441 
.5381452 
032462 
533471 
5384478 
535485 
. 586490 
537495 


8.538498 
.539501 
540502 
541502 
.542501 
543499 
544497 
545493 
546488 

8.547482 





Di 


OT OT Sd D3 SSD 
C2 2 00 © Oo Co = 


bm ek ek ed ee 


EAP WPAPHP PI YrF I-VI 


me BL ororor 
CarRMmowe 


WII 
oS 
ri 


AFI 
~~ 
ao 


_ 
oS 
ile) 
[e2) 









































15° 
y Vers, | D. 1’. | Ex. sec, Dr iy: 
0 "3. 532425 15.98 (8.547482 16: 53 
1 .538884 | 15.97 | .548474 | 16.53 
2 534342 | 15.95 NG | 16.52 
a 535299 | 15.93 | .550457 | 16.50 
4 536255 | 15.92 | .551447 | 16.48 
5 537210 | 15.88 | .552486 | 16.47 
6 588163 | 15.88 | .553424 | 16.43 
539116 | 15.87 | .554410 ! 16.43 
8 540068 | 15.838 | .555396 | 16.42 
9 541018 | 15.83 | .556381 | 16.38 
10 541968 | 15.80 | .557364 | 16.38 
11 | 8.542916 | 15.78 |8.558347 | 16.37 
12 .548863 | 15.78 | .559329 | 16.33 
13 .544810 | 15.7 .5603809 | 16.33 
14 45755 | 16.7: -561289 | 16.30 
15 .546699 | 15.7% .562267 | 16.30 
16 -547642 | 15.7 .563245 | 16.28 
17 .548584 | 15.68 | .564222 | 16.25 
18 -549525 | 15.67 | .565197 | 16.25 
19 .550465 | 15.65 | .566172 | 16.22 
20°) ,551404 | 15.63 |..567145 | 16.22 
21 | 8.552342 | 15.62 |8.568118 | 16.20 
22 .5538279 -| 15.60 | .569090 | 16.17 
23 -654215 | 15.58 | .570060 | 16.17 
24 .555150 | 15.57 | .571080 | 16.15 - 
2a .556084 | 15.55 | .571999 | 16.12 
26 557017 | 15.53 | .572966 | 16.12 
27 .557949 | 15.50 | .5738983 | 16.10 
28 .558879 | 15.50 | .574899 | 16.08 
29 -559809 | 15.48 | .575864 | 16.05 
80 | .560788 | 15.47 | .576827 | 16.05 
81 | 8.561666 | 15.48 |8.577790 | 16.08 
82 .662592 | 15.48 | 578752 | 16.02 
83 .563518 | 15. 42 579713 | 16.00 
34 .564448 | 15.40 | .580673 | 15.98 
85 .565367 | 15.387 | .581682 | 15.97 
86 .566289 | 15.87 | .582590 | 15.95 
37 -567211 | 15.35 | .588547 | 15.98 
38 -5681382 | 15.33 | .584503 | 15.92 
39 -569052 | 15.80] .585458 | 15.90 
40 -5669970 | 15.80 | .586412 | 15.88 
41 | 8.570888 | 15.28 8.587365 | 15.88 
42 .571805 | 15.27 | .588818 | 15.85 
43 .572721 | 15.25 | .589269 | 15.83 
44 .573636 | 15.22 | .590219 | 15.83 
45 .574549 | 15.22 | .591169 | 15.80° 
46 .575462 | 15.20 | .592117 | 15.80 
47 .576374 | 15.18 | .598065 | 15.78 
' 48 .577285 | 15.17 | .594012 | 15.%5 
49 .578195 | 15.15 | .594957 | 15.75 
50 .579104 | 15.13 | .595902 | 15.78 
51 | 8.580012 | 15.12 |8.596846 | 15.7 
52! .580919 | 15.10 ! .597°789 | 15.70 
53 .581825 | 15.08 | .5987381 | 15.68 
54 .582730 | 15.07 | .599672 | 15.67 
bb .588634 | 15.05 | .600612 | 15.65 
56 .5845387 | 15.05 | .601551 | 15.65 
57 .585440 | 15.02 | .602490 | 15.62 | 
58 .586341 | 15.00 | .608427 | 15.60 
59 | ,587241 | 15.00 | .604363 5.60 
60 | 8.588141 | 14.97 |8.605299 | 15.58 - 


AND EXTERNAL SECANTS 


143 





es | ee | | | | 





, Vers. 
0 | 8.588141 
1 .589039 
2 589936 
3 590833 
4 591729 
5 592623 
6 593517 
vg 594410 
8 .595302 
g .596192 
10 .597082 
11 | 8.597971 
12 .598860 
13-| .599747 
14 . 600633 | 
15 .601518 
16 .602403 
17 . 603286 
18 .604169 
19 . 605051 
20 .605931 
21 | 8.606811 
22 .607690 
23 | .608568 
24 609445 
25 .610321 
26 .611197 
27.) .612071 
98 .612945 
29 2613817 
30 | .614689 
31 | 8.615560 
32 . 616430 
33 .617299 
84 | .618167 
35 .619034 
36 .619901 
ot .620766 
38 . 621631 
39 . 622495 
40 6238358 
41 | 8.624220 
42 . 625081 
43 . 625941 
44 .626801 
45 -627659 
46 628517 
47 . 629374 
48 .630230 
49 .631085 
50; .631939 
51 | 8.632792 
52 . 633645 
53 634496 
54 . 635347 
55 -636197 
56 .637046 
57 637894 
58 .638742 
59 . 639588 
60 | 8.640434 











16° 

D.}".1| Ex..sec. 
14.97 | 8.605299 
14,95 . 606234 
14.95 , 607167 
14.93 .608100 
14.90 .609032 
14.90 .609963 
14.88 .610893 
14.87 611823 
14.83 .612751 
14.83 . 613678 
14.82 | .614605 
14.82 | 8.615531 
14.78 .616456 
14.77 .617379 
14.75 .618302 
14.75 . 619225 
14:72 .620146 
14.72 .621066 
14.70 .621986 
14.67 . 622904 
14.67 623822 
14.65 | 8.624739 
14.63} .625655 
14.62 . 626570 
14.60 - 627484 
14.60 .628398 
14.57 .629310 
14.57 . 630222 
14.53 .6311383 
14.53 . 6320438 
14.52 | . .632952 
14.50 | 8.633860 
14.48 .634768 
14.47 .635674 | 
14.45 .636580 
14.45 . 637485 
14.42 . 638389 
14.42 .639292 
14.40 .640195 
14.38 .641096 
14.37 .641997 
14.35 | 8.642897 
14.33 .643796 
14.33 .644694 
14.30 . 645591 
14.30 .646488 
14.28 647384 
14.27 .648279 
14.25 .649173 
14.23 . 650066 
14.22 | .650958 
14.22 | 8.651850 
14.18 652741 
14.18 1. .653631 
14.17 654520 
14,15 655408 
14.13 656296 
14.13 657182 
14.10 658068 
14.10 658954" 
14.08 | 8.659838 























WHWHWNMWNWNHWWHKW 
SO Dk So OU CO 0 
I 


Vers. 





"8.640434 


-641279 
-642123 
642966 
.643809 
-644650 
-645491 
.646331 
.647170 
-648008 
-648845 


8.649682 
.650518 
.651353 
.652187 
653020 
653852 
654684 
.655515 
-656345 
657174 


8.658003 
.658830 
-659657 
-660483 
.661308 
-662132 
-662956 
663779 
-664601 
665422 


8.666242 
. 667062 
667881 
. 668699 
.669516 
-670332 
.671148 
.671963 
672777 
.673590 

8.674403 
675215 
. 676026 
676836 
677645 
678454 
679262 
.680069 
.680875 
.681681 


8.682486 
683290 
-684093 
684896 
685697 
686498 
“687299 
688098 
688897 

8.689695 | 


17° 


14.07 | .660721 
14.05 | .661604 
14.05 | .662486 
14.02 | .663367 
14.02 | .664248 
14.00 | .665127 
13.98 | .666006 
13.97 | .666884 


13.95 | .667761 
13.95 | .668637 


13.93 |8.669513 


13.92 | .670388 
13.90 | .671262 
13.88 | .672135 
13.87 | .673008 


13.87 | .673879 
18.85 | .674750 
13.83 | .675620 
13.82 | .676490 
13.82 | .677358 


13.73 | .682554 
13.72 | .6838417 
13.70 | .684280 
18.68 | .685141 
13.67 | .686002 
13.67 8.686863 


13.62 | .689439 
13.66 | .690296 
13.60 | .691153 
13.58 | .692008 


13.45 | .700525 
13.43 | .701373 
13.43 | .702220 





13.40 |8.703912 
13.38 | .704756 


13.38 | .’705600 
13.35 | .706444 
13.35 | .707286 


13.35 | .708128 
13.82 | .708969 
13.82 | .709810 
13.30 | .710650 
13.28 |8.711489 





D. 1”, | Ex. ea 0 


14.08 |8.659838 


14.72 © 
14.72 
14.70 
14.68 
14.68 
14.65 
14.65 
14.68 
14.62 
14.60 
14.60 


14.58 
14.57 


errr RUS err no I I TE SE Oa FET 


144 TABLE XXVI.—LOGARITHMIC VERSED SINES 
————S ee 

































































18° 19° 
Sat ie ge ht st onc ah 
U Vers. | D. 1". | Ex. sec. | D. 1". U Vers. D. 1". | Ex. sec./D. 1” 
0 | 8.689695 | 138.28 | 8.711489 | 13.97 0 | 8.736248 | 12.58 |8.760578 | 13.30 
1 .690492 | 13.28 712327 | 18.95 1 . 737008 | 12.57 | .7613876 | 13.30 
2 .691289 | 18.25 - 7138164 | 13.95 2 U37757 | 12.55 | 762174 | 13.28 
3 .692084 | 138.25 .714001 | 138.95 3 738510 | 12.55 | .762971 | 13.97 
4 .692879 | 13.25 .714888 | 13.92 4 739263 | 12.53 | .768767 | 13.97 
5 .693674 | 13.22 . 715678 | 13.92 5 -740015 | 12.52 | .764563 | 13 25 
6 .694467 | 13.22 -716508 | 13.90 6 -740766 | 12.50 | .765358 | 13.23 
Mi .695260 | 13.20 .7173842 | 13.88 7 -741516 | 12.50 | .766152 | 13.23 
8 .696052 | 13.18 .718175 | 13.88 8 - 742266 | 12.50 | .766946 | 13.22 
9 .696848 |. 13.18 -719008 | 13.87 9 - 743016 | 12.47 | .767739 | 13.20 
10 .697634 | 13.17 . 719840 | 18.85 || 10 - 743764 | 12.47 | .768531 | 13.20 
11 | 8.698424 | 13.15 | 8.720671 | 13.85 || 11 8.744512 | 12.45 |8.'7693823 | 13.18 
12 .699213 | 13.13 . 721502 | 13.88 12 . 745259 | 12.43 | .770114 | 13.18 
ats .700001 | 13.13 . 722882 | 13.82 || 13 -746006 | 12.45 | .770905 | 13.17 
14 .700789 | 13.12 . 723161 | 13.80 || 14 (46752 | 12.42 | .771695 | 13.15 
15 _ 701576 | 13.10 . 723989 | 13.80 || 15 ~T47497 | 12.42 | .772484 | 18.15 
16 . 702862 | 13.08 - 724817 | 18.78 || 16 (48242 | 12.40 | .773273 | 13.13 
17 . 708147 | 18.08 725644 | 18.7 17 -748986 | 12.388 | .774061 | 13.13 
18 703932 | 13.07 -726471 | 18.77 || 18 - 749729 | 12.38 | .774849 | 13.12 
19 - 704716 | 138.05 ALS By day HL SKS) . 750472 | 12.37 | .775636 | 13.10 
20 -705499 | 13.05 728122 | 138.7 20 -751214 | 12.35 | .776422 | 13.08 
21 | 8.706282 | 13.02 | 8.728946 | 13.73 || 21 | 8.751955 | 12.35 8.777207 | 13.10 
22 . 707063 | 13.02 729770 | 13.72 || 29 . 752696 | 12.33 | .777993 | 13.07 
23 . 707844 | 13.02 730593 | 13.70 || 23 758486 | 12.32 | .778777 | 13.07 
24 - 708625 | 12.98 - (381415 | 13,70 |) 24 -T54175 | 12.382 |. .779561 | 13.05 
25 . 709404 | 12.98 - 782237 | 13.68 || 25 .754914 | 12.30 | .780844 | 13.05 
26 .710183 | 12.98 . 738058 | 13.67 || 26 755652 | 12.28 | .781127 | 13.03 
27 . 710961 | 12.95 . 7383878 | 13.67 ij . 756889 | 12.28 | .781909 | 13.02 
28 .711739 | 12.95 734698 | 13.65 || 28 757126 | 12.27 | .782690 | 18.02 
29 .712516 | 12.93 . 785517 | 13.68 || 29 757862 | 12.27 | .783471 | 13.00 
30 - 718292 | 12.92 . 786385 | 13.68 || 80 758598 | 12.25 | .784251 | 13.00 — 
31 | 8.714067 | 12.92 | 8.737153 | 13.62 || 31 | 8.'759393 12.23 8.785031 | 12.98 
32 . 714842 | 12.90 . 737970 | 13.60 || 82 760067 | 12.23 | .785810 | 12.97 
33 . 715616 | 12.88 . 738786 | 18.60 || 33 . 760801 | 12.22 | .786588 | 12.97 
34 . 716389 | 12.87 - 739602 | 13.58 || 34 .761534 | 12.20 | .787366 | 12.97 
35 .717161 | 12.87 - 740417 | 18.57:1) 85 762266 | 12.20 | .788144 | 12.93 
36 717983 | 12.85 . 741231 | 18.57 || 86 . 762998 | 12.18 | .788920 | 12.93 
37 - 718704 | 12.85 . 742045 | 138.55 || 37 763729 | 12.17 | .789696 | 12.93 
38 7719475 | 12.82 - 742858 | 13.53 || 38 764459 | 12.17 | .790472 | 12.92 
39 . 720244 | 12.82 - 743670 | 13.53 || 39 .(65189 | 12.15 | .791247 | 12.90 
40 . 721018 | 12.82 - 744482 | 18.52 || 40 -(65918 | 12.15 | .792021 | 12.90 
41 | 8.721782 | 12.78 | 8.745293 | 13.50 || 41 | 8.766647 12.12 |8.792795 | 12.88 
42 . 722549 | 12.7 . 746108 | 18.50 || 42 . 767374 | 12.12 | .793568 | 12.87 
43 (20016 | 1227 . 746913 | 18.48 || 43 . 768102 | 12.10 | .794340 | 12.87 
44 . 724083 | 12.75 - 747722 | 18.47 || 44 . 768828 | 12.10 | .795112 | 12.87 
45 . 724848 | 12.7: . 748580 | 13.47 || 45 769554 | 12.10 | .795884 | 12.83 
46 feoole | 12773 . 749338 | 13.45 || 46 770280 | 12.08 | .796654 | 12.85 
47 126877 | 12272 - 750145 | 18.43 || 47 771005 | 12.07 | .797425 | 12.82 
48 727140 | 12.7 750951 | 13.48 || 48 -771729 | 12.05 | .798194 | 12.82 
49 .727903 | 12.7 wolfe? | 13042 |1049 772452 | 12.05 | .798963 | 12.82 
50 728665 | 12.7 . 752562 | 18.42 || 50 -773175 | 12:05 | .799732 | 12.80 
51 | 8.729427 | 12.67 | 8.753367 | 13.40 || 51 8.773898 | 12.02 |8.800500 | 12.78 
52 . 730187 | 12.67 . 754171 | 13.38 || 52 -774619 | 12.02 | .801267 | 12.7 
53 . 730947 | 12.67) .754974 | 13.37 || 53 . 775340 | 12.02 | .802034 |} 12.77 
54 . 731707 | 12.63 755776 | 13.87 || 54 . 776061 | 12.00 | .802800 | 12.75 
BD . 782465 | 12.63 .756578 | 18.387 || 55 .776781 | 11.98 | .803565 | 12.75 
56 . 7338223 | 12.63 757380 | 13.33 || 56 .777500 | 11.97 | .804330 | 12.75 
57 . 733981 | 12.60 - 758180 | 18.33 || 57 : 778218 11.97 | .805095 | 12.73 
58 . 784737 | 12.60 . 758980 | 13.33 || 58 .778936 | 11.97 | .805859 | 12.72 
59 . 735493 | 12.58 . 759780 | 13.30 || 59 79654 11.93 | .806622 | 12.72. 
60 | 8.736248 | 12.58 | 8.760578 | 13.30 || 60 | 8.780370 | 11.95 8.807385 | 12.70 | 
ES SS 


AND EXTERNAL SECANTS 


ees 





























20° 
4 Vers... | D. 1". | Ex. sec. | D. 1”. 
0 | 8.780870 | 11.95 | 8.807385 | 12.70 
1 .781087 | 11.92 .808147 | 12.68 
2 .781802 | 11.92 .808908 | 12.68 
3 .782517 | 11.90 .809669 | 12.68 
4 . 783231 | 1f.90 .810480 | 12.67 
5 .783945 | 11.88 .811190 | 12.65 
6 784658 | 11.88 .811949 | 12.65 
a 785371 | 11.87 .812708 | 12.63 
8 . 786083 | 11.85 .813466 | 12.63 
9 . 786794 | 11.85 .814224 | 12.62 
40 787505 | 11.88 .814981 | 12.60 
41 | 8.788215 | 11.82 | 8.815737 | 12.60 
12 .788924 | 11.82 .816493 | 12.60 
13 . 789633 | 11.82 ~817249 | 12.58 
14 .790342 | 11.78 .818004 | 12.57 
15 .791049 | 11.78 .818758 | 12.57 
16 .791756 | 11.78 .819512 | 12.55 
17 792463 | 11.77 . 820265 | 12.55 
18 .793169 | 11.75 .821018 | 12.53 
19 . 7938874. | 11.75 .821770 | 12.52 
20 .794579 | 11.78 822521" | 12.52 
91 | 8.795283 | 11.73 | 8.823272 | 12.52 
22 .795987 | 11.7% .824023 | 12.50 
Qe . 796690 | 11.70 824773 | 12.48 
24 .797392 | 11.7 . ,825522 | 12.48 
25 .798094 | 11.68 7826271 | 12.47 
26 .798795 | 11.68 827019 | 12.47 
27 .799496 | 11.67 .827767 | 12.45 
2 .800196 | 11.67 .828514 | 12.45 
29 .800896 | 11.63 .829261 | 12.43 
30 .801594 | 11.55 .830007 | 12.42 
31 | 8.802293 | 11.63 | 8.820752 | 12.42 
32 .802991 | 11.62 .831497 | 12.42 
33 .803688 | 11.60 832242 | 12.40 
34 .804384 | 11.60 .832986 | 12.38 
oo .805080 | 11.60 .833729 | 12.38 
36 805776 | 11.58 834472 | 12.388 
387 .806471 | 11.57 ,835215 | 12.37 
38 807165 | 11.57 .835957 | 12.35 
39 .807859 | 11.55 .836698 | 12.35 
40 .808552 | 11.53 .837439 | 12.33 
41 | 8.809244 | 11.53 | 8.888179 | 12.38 
42 -809936 | 11.53 .838919 | 12.32 
43 .810628 | 11.52 .839658 | 12.30 
44 .8113819 | 11.50 .840396 | 12.32 
45 .812009 | 11.50 .841135 | 12.28 
46 .812699 | 11.48 .841872 | 12.2 
47 .813388 |- 11.48 .842609 | 12.28 
48 .814077 | 11.47 .843346 | 12.27 
49 .814765 | 11.45 .844082 | 12.25 
50 .815452 | 11.45 .844817 | 12.25 
51 | 8.816139 | 11.43 | 8.845552 | 12.25 
52 .816825 | 11.43 .846287 | 12.23 
53 817511 | 11.42 .847021 | 12.22 
54 .818196 | 11.42 .847754 | 12.22 
Do .818881 | 11.40 848487 | 12.22 
56 .819565 | 11.40 .849220 | 12.20 
57 .820249 | 11.38 ,849952 | 12.18 
58 .820932 | 11.37 .850683 | 12.18 
59 .821614 | 11.37 .851414 | 12.17 
60 | 8.822296 | 11.35 | 8.852144 














Zi9 
, Vers. D. 1’. | Ex. sec.|D. 1°. 
0 | 8.822296 | 11.35 |8.852144 | 12.17 
1 ,822977 | 11.35 | .852874 | 12.17 
2 ,823658 | 11.33 | .853604 | 12.13 
3 .824338 | 11.88 | .85438382 | 12.15 
4 /825018 | 11.32 | .855061 | 12.13 
5 .825697 | 11.32 | .855789 | 12.12 
6 826876 | 11.380 | .856516 | 12.12 
7 ~827054 | 11.28 | .857243 | 12.10 
8 827731 | 11.28 | .857969 | 12.10 
9 .828408 | 11.28 | .858695 | 12.08 
10 .829085 | 11.27 | .859420 | 12.08 
11 | 8.829761 | 11.25 |8.860145 | 12.07 
12 ,830436 | 11.25 | .860869 | 12.07 
13 .831111 | 11.23 | .861598 | 12.05 
14 .831785 | 11.23 | .862816 | 12.05 | 
15 832459 | 11.22 | .863039 | 12.03 
16 ,833132 | 11.20 | .8638761 | 12.03 
17 -833804 | 11.20 | .864483 | 12.02 
18 834476 | 11.20 | .865204 | 12.02 
19 -835148 | 11.18 | .865925 | 12.02 
20 6835819 | 11.17 | .866646 | 11.98 
21 | 8.836489 | 11.17 |8.867365 | 12.00 
22 837159 | 11.17 | .868085 | 11.98 
23 837829 | 11.15 | .868804 | 11.97 
24 838498 | 11.13 | .869522 | 11.97 
25 839166 | 11.13 | .870240 | 11.95 
26 .839834 | 11.12 | .870957 | 11.95 
Pt 840501 | 11.12 | .871674 | 11.93 
28 .841168 | 11.10 | .872390 | 11.93 
29 841834 | 11.10 | .873106 | 11.93 
30 842500 | 11.08 | .878822 | 11.92 
31 | 8.843165 | 11.07 |8.874537 | 11.90 
382 843829 | 11.07 | .875251 | 11.90 
33 844498 | 11.07 | .875965 | 11.88 
34 845157 | 11.05 | .876678 | 11.88 
35 845820 | 11.05 | .877391 | 11.88 
36 846483 | 11.03 | .878104 | 11.87 
37 847145 | 11.02 | .878816 | 11.87 
38 847806 | 11.02 | .879528 | 11.85 
39 848467 | 11.00 | .880239 | 11.83 
40 849127 | 11.00 | .880949 | 11.83 
41 | 8.849787 | 11.00 |8.881659 | 11.88 
42 850447 | 10.98 | .882869 | 11.82 
43 851106 | 10.97 | .8838078 | 11.82 
44 851764 | 10.97 | .8838787 | 11.80 
45 852422 | 10.95 | .884495 | 11. 80 
46 853079 | 10.95 | .885203 | 11.78 
47 853736 | 10.93 | .885910 | 11.7 
48 854392 | 10.93 | .886617 | 11.77 
49 855048 | 10.92 | .887823 | 11.77 
50 855703 | 10.92 | .888029 | 11.75 
51 | 8.856358 | 10.90 |8.888734 | 11.75 
52 857012 | 10.90 | .889489 | 11.75 
53 857666 | 10.88 | .890144 | 11.7 3 
54 858319 | 10.88 | .890848 | 11.72 
55 858972 | 10.87 | .891551 | 11.72 
56 859624 | 10.87 | .892254 | 11 72. 
57 .860276 | 10.85 | .892957 | 11 70 
58 860927 | 10.85 | .893659 | 11 70 
59 861578 | 10.83 | .894361 | 11.68 
60 | 8.862228 | 10.82 |8.895062 | 11.68 
































2 Ud atta EIT ih Td a a i ee A ae 


146 TABLE XXVI.—LOGARITHMIC VERSED SINES 
ey 


22° 23° 





~ 


Vers? 1D: or Exesee: | Del” Vers. } D. 1’. | Ex. sec.'D. 1°. 





8.900341 | 10.33 (8.936315 | 11.23 
.900961 | 10 35 | .986989 | 11.23 





, 
8.862228 | 10.82 | 8.895062 | 11.68 0 
862877 | 10.83 | .895763 | 11.67 i 
863527 | 10.80 | .896463 | 11.67 2] .901582 | 10.82 | .987663 | 11.22 
.864175 | 10.80 | .897163 | 11.65 3 | .902201 | 10.33 | .988336 | 11.22 
.864823 | 10.80 | .897862 | 11.65 2 -902821 | 10.32 | .939009 | 11.22 
6 
7 
re) 
9 











: : .898561 | 11.63 903440 ; 10.30 | .939682 | 11.20 
.866118 | 10.7 899259 | 11.63 904058 | 10.30 | .940354 | 11 20 
.866765 | 10.77 | .899957 | 11.63 -904676 | 10.28 | .941026 | 11.20 
867411 | 10.77 | .900655 | 11.62 .905293 | 10.28 | .941698 | 11.18 
868057 | 10.75 .901352 | 11.60 -905910 | 10.28 | .942369 | 11 17 
-868702 | 10.73 | .902048 | 11.62 || 10 | .906527 | 10.27 | .943039 | 11.18 


11 | 8.869346 | 10.75 | 8.902745 | 11.58 |} 11 | 8.907143 | 10.27 |8.943710 | 11 15 
12 869991 | 10.7% -903440 | 11.60 || 12] .907759 | 10.25_| .944879 | 11.17 
13 | .870634 | 10.72 | .904136 | 11.57 || 13 .908374 | 10.25 | .945049 | 11.15 
14 871277 | 10.7: -904830 | 11.58 || 14 -908989 | 10.23 | .945718 | 11.13 
15 | .871920 | 10.7 905525 | 11.57 || 15 | .909603 | 10.23 | .946386 | 11.13 
16 872562 | 10.70} .906219 | 11.55 || 16 910217 | 10.22 | .947054 | 11.13 
17 . 873204 | 10.68 | .906912 | 11.55 || 17 | .910830 | 10.22 | .947722 | 11 12 
18 .873845 | 10.68 | .907605 | 11.53 |} 18 | .911443 | 10.22 | .948389 | 11.12 
19 874486 | 10.67 | .908298 | 11.53 |] 19 | .912056 | 10.20 | .949056 | 11.12 
20 | .875126 | 10.67 | .908990 | 11.52 /| 20 | .912668 | 10.18 | .949723 | 11.10 


21 | 8.875766 | 10.65 | 8.909681 | 11.52 || 21 | 8.913279 | 10.20 8.950889 | 11.10 
22 .876405 | 10.65 | .910372 | 11.52 || 22 | .9138891 | 10.17 | -.951055 | 11.08 
23 | .877044 | 10.63 | .911063 | 11.52 || 23 914501 | 10.17 | .951720 | 11.08 
24 877682 | 10.63 | .911754 | 11.48 || 24 915111 | 10.17 | .952885 | 11.07 
25 .878320 | 10.62 | .912443 | 11.50 || 25} .915721 | 10.17 | .953049 | 11.07 
26 | .878957 | 10.62 | 913133 | 11.48 |} 26 | .916331 | 10.15 | .953713 | 11.07 
27 | .879594 | 10.60 | .913822 | 11.47 |} 27 | .916940 | 10.13 | .954877 | 11.05 
28 -880230 | 10.60 | .914510 | 11.47 || 28} .917548 | 10.13 | .955040 | 11.05 
29 | .880866 | 10.58 -915198 | 11.47 || 29 | .918156 | 10.13 | .955703 | 11.05 
30 | .881501 | 10.58 | .915886 | 11.45 || 30 .918764 | 10.12 | .956366 | 11.03 


31 | 8.882136 | 10.58 | 8.916573 | 11.45 || 31 | 8.919371 | 10.10 |8.957028 | 11.03 
32 | 882771 | 10.57 | .917260 | 11.43 || 32 | .919977 | 10.12 | .957690 | 11 02 
33 | .883405 | 10.55 917946 | 11.43 || 833} .920584 | 10.10 | .958351 | 11.02 
34 . 884038 | 10.55 | .918682 | 11.48 || 34 921190 | 10.08 | .959012 | 11.00 
35 .884671 | 10.53 | .919318 | 11.42'|| 35 | .921795 | 10.08 | .959672 | 11.00 
36 .885303 | 10.53 | .920003 | 11.40 |} 36 | .922400 | 10.07 | .960382 | 11.00 
37 -885935 | 10.53 | .920687 | 11.42 || 87} .923004 | 10.07 | .96°992 | 10.98 
38, .886567 | 10.52 | .921372 | 11.88 || 88 | .923608 | 10.07 | .981651 | 10.98 
39 | .887198 ( 10.52 |  .922055 | 11.40 || 39 | .924212 | 10.05 | .969310 | 10.98 
40 | .887829 | 10.50 | .922789 | 11.387 || 40 | .924815 | 10.05 | .962969 | 10.97 


41 | 8.888459 | 10.48 | 8.923421 | 11.38 || 41 | 8.925418 | 10.08 8.963627 | 10.97 
42-; .889088 | 10.48 | .924104 | 11.37 || 42 | .926020 | 10.03 | .964985 | 10.95 
43} .889717 | 10.48 | .924786 | 11.35 || 43 | 926622 | 10.03 | .964942 | 10.95 
44 | .890346 | 10.47 | 925467 | 11.37 || 44] 927224 | 10.02 | .965599 | 10.95 
45 | .890974 | 10.47 | .926149 | 11.33 || 45 | .927825 | 10.00 | .966256 | 10.93 
46 | .891602 | 10.45 |- 926829 | 11.35 || 46 | .928425 | 10.00 | .966912 | 10.93 
47 |  .892229 | 10.45 | .927510 | 11.32 || 47 | .929025 | 10.00 | .967568 | 10.92 
48 | .892856 | 10.43 | .928189 | 11.33 || 48 | .929625 | 9.98 | .968223 | 10.92 
49 | .893482 | 10.43 .928869 | 11.382 || 49 | .930224 | 9.98 | .968878 | 10.92 
50 | .894108 | 10.42 | .929548 | 11.30 || 50 | .930828 | 9.97 | .969533 | 10.90 


51 | 8.894733 | 10.42 | 8.930226 | 11.82 || 51 | 8.931421 | 9.97 [8.970187 | 10.90 
52 .895358 | 10.42 | .930905 | 11.28 || 52 | .932019 | 9.97] .970841 | 10.88 
53 ,895983 | 10.40 | .931582 | 11.30 || 53 | .932617 | 9.95 | .971494 | 10.88 
54 896607 | 10.38 | .932260 | 11.27 || 54 | .933214 | 9.95 |. .972147 | 10.88 
55; .897230 | 10.38 | .932936 | 11.28 || 55 | .933811 | 9.93 | .972800 | 10 87 
56 | .897853 | 10.38 | .933613 | 11.27 || 56 | .934407 | 9.93 | .973452 | 10°87 
57 | .898476 | 10.37 | .934289 | 11.27 || 57 | .935003 | 9.92 | .974104 | 10.87 
58 | .899098 | 10.35 | .934965 | 11.25 | 58 | .935598 eee 974756 | 10.85 
59) .899719 | 10.37 | .935640 | 11.25 || 59 | .936193 | 9.92 | .975407 | 10.85 
60 | 8.900341 | 10.33 | 8.936315 | 11.23 || 60 | 8.936788 | 9. 90 8.976058 | 10.83 


Eee ee 


i" 
SUM OMIAOURWWH SO 
le 2) 

St 
rs 
J 
_ 

i. 

oO 
~} 

(es) 









































AND EXTERNAL SECANTS 147 







































































24° 25° 

, Vers. [{D.1".} Exssee. | D. 17, [iP Vers: | D1". Ex. gee:|D. as 
0 | 8.986788 9.90 | 8.976058 | 10.83 0 | 8.971703 | 9.50 | 9.014428 | 10.47 
1 . 987382 9.90 .976708 | 10.83 1 .972273 | 9.48 .015056 | 10.48 
2 . 937976 9.88 977358 | 10.88 2 .972842 | 9.48 .015685 | 10.45 
3 . 98855 9.88 .978008 | 10.82 3 .973411 | 9.48 .016312 | 10.47 
4 .989162 9.87 .978657 | 10.82 4 .973980 | 9.47 .016940 | 10.45 
5 939754 9.87 979306 | 10.80 5 974548 | 9.47 .017567 | 10.45 
6 940346 9.87 979954 | 10.80 6 .975116 | 9.45 .018194 | 10.45 
"6 940938 9.85 980602 | 10.80 y | .9756838 | 9.45 .018821 | 10.43 
8 941529 9.85 981250 | 10.80 8 -976250 | 9.438 .019447 | 10.43 
9 . 942120 9.83 .981898 | 10.7 9 .976816 | 9.43 .020073 | 10.42 
10 . 942710 9.83 .982545 | 10.77 || 10 | + .977382 | 9.48 .020698 | 10.42 
11 | 8.943300 9.82 | 8.983191 | 10.77 || 11 | 8.977948 | 9.43 | 9.021323 | 10.42 
12 . 948889 9.83 .9838837 | 10.77 12 .978514 | 9.42 .021948 | 10.40 
13 . 944479 9.80 .984483 | 10.77 |} 13 .979079 | 9.40 .022572 | 10.42 
14 - 945067 9.80 .985129 | 10.75 || 14 .979648 | 9.40 -023197 | 10.38 
15 . 945655" | 9.80 . 98577 10.75 15 - 980207 | 9.40 .023820 | 10.40 
16 . 9462438 9.80 .986419 | 10.7 16 .980771 | 9.40 .024444 | 10.38 
17 . 946831 9.78 .987063 | 10.73 || 17 .981385 | 9.38 .025067 | 10.38 
18 . 947418 Oats. .987707 | 10.7 18 -981898 | 9.37 -025690 | 10.37 
19 . 948004 9.77 .988351 | 10.72 |} 19 . 982460 | 9.38 .026312 | 10.37 
20 . 948590 TANG .988994 | 10.72 || 20 - 98380238 | 9.37 .026934 | 10.37 
21 | 8.949176 9.75 | 8.989637 | 10.7 21 | 8.983585 | 9.85 | 9.027556 | 10.35 
22 . 949761 9.75- 990279 | 10.7 22 .984146 | 9.35 028177 | 10.35 
23 . 950346 | 9.75 . 990922 | 10.68 |} 23 .984707 | 9.35 028798 | 10.35 
24 . 9509381 9.73 .991563 | 10.70 |} 24 . 985268 | 9.33 .029419 | 10.38 
25 .951515 9.73 . 992205 | 10.68 |} 25 .985828 | 9.33 .080089 | 10.33 
26 . 952099 9.72 . 992846 | 10.68 || 26 - 986385 | 9.33 -030659 | 10.38 
27 . 952682 9.72 .998487 | 10.67 || 27 . 986948 | 9.32 031279 | 10.338 
28 . 9538265 9.70 .994127 | 10.67 || 28 -987507 | 9.32 031899 | 10 32 
29 . 953847 9.70 .994767 | 10.65 || 29 . 988066 | 9.32 032518 | 10.30 
30 . 954429 9.70 .995406 | 10.67 || 30 .988625 | 9.30 033186 | 10.28 
81 | 8.955011 9.68 | 8.996046 | 10.65 || 31 | 8.989183 | 9.28 |} 9.033755 | 10.30 
32 . 955592 9.68 .996685 | 10.63 |} 32 .989740 | 9.30 .034373 | 10.30 
33 . 956173 9.67 .997323 | 10.63 |} 33 . 990298 | 9.28 .0384991 | 10.28 
34 . 956758 9.68 .997961 | 10.63 || 34 -990855 | 9.27 .035608 | 10.28 
35 . 957384 9.65 .998599 | 10.62 || 35 991411 | 9.28 .036225 | 10.28 
36 . 957913 9.65 .9992386 | 10.62 || 36 .991968 | 9.25 036842 | 10.27 
ov |. .958492 9.65 | 8.999873 | 10.62 || 87 . 9925238 | 9.27 0387458 | 10.27 
38 . 959071 9.65 | 9.000510 | 10.60 |} 38 .993079 | 9.25 038074 | 10.27 
39 . 959650 9.63 .001146 | 10.62 || 39 .993634 | 9.25 038690 | 10.25 
40 | . 960228 9.62 001783 | 10.58 || 40 .994189 | 9.23 039805 | 10.25 
41 | 8.960805 9.62 | 9.002418 | 10.58 || 41 | 8.994743 | 9.23 | 9.039920 | 10.25 
42 . 961382 9.62 .003053 | 10.58 || 42 - 995297 | 9.23 040535 | 10.25 
43 . 961959 9.60 .003688 | 10.58 || 43 .995851 | 9.22 041150 | 10.23 
44 . 962585 $.60 .004823 | 10.57 || 44 . 996404 | 9.22 .041764 | 10.23 
45 . 9638111 9.60 .004957 | 10.57 || 45 . 996957 | 9.20 .042378 | 10.22 
46 . 963687 9.58 .005591 | 10.55 || 46 997509 | 9.22 .042991 | 10.22 
47 . 964262 9.58 .006224 | 10.57 || 47 .998062 | 9.18 .043604 | 10.22 
48 . 964837 9.57 .006858 | 10.53 || 48 .998613 | 9.20 044217 | 10.22 
49 .965411 9.57 .007490 | 10.55 || 49 .999165 | 9.18 .044830 | 10.20 
50} .965985 9.57 .008123 | 10.53 || 50 | 8.999716 | 9.17 -045442 | 10.20 
51 | 8.966559 9.55 | 9.008755 | 10.53. || 51 | 9.000266 | 9.18 | 9.046054 | 10.18 
52 . 967132 9.55 *009387 | 10.52 || 52 .000817 | 9.17 .046665 | 10.18 
53 . 967705 9.53 .010018 | 10.52 || 53 .001867 | 9 15 047276 | 10.18 
54 968277 9.53 | * .010649 | 10.52 || 54 .001916 | 9.17 .047887 | 10.18 
55 968849 9.53 .011280 | 10.50 || 55 .002466 | 9.13 .048498 | 10.17 
56 . 969421 9.52 .011910 | 10.50 || 56 .008014 | 9.15 .049108 | 10.17 
57 . 969992 9.52 .012540 | 10.50 || 57 .003563 | 9.13 049718 | 10.17 
58 . 970563 9.50 .013170 | 10.48 || 58 .004111 | 9.13 .050828 | 10.15 
59 .971133 9.50 .013799 | 10.48 || 59 .004659 | 9.12 .050937 | 10.15 
60 | 8.971703 9.50 | 9.014428 | 10.47 || 60 | 9.005206 | 9.12 | 9.051546 | 10.15 
a ee 


148 TABLE XXVI.—LOGARITHMIC VERSED 


SINES 


——— Cho ll ES 











26° 

/ Vers.4a-D. i". 
0 | 9.005206 9.12 

1 .005753 | 9.12 
2 .006300 9.10 
3 .006846 9.10 
4 .007392 | 9.10 
5 .007938 | 9.08 
6 .008483 9.08 
& .009028 | 9.07 
8 .009572 | 9.07 
9 .010116 9.07 
10 .010660 | 9.05 
11 | 9.011203 9.05 
12 .011746 9.05 
13 .012289 9.03 
14 .012831 9.03 
15 .013373 | 9.08 
16 .013915 | 9.02 
17 .014456 | 9.02 
18 .014997 | 9.02 
19 .015538 | 9.00 
20 .016078 9.00 
21 | 9.016618 | 8.98 
22 -017157 | 9.00 
23 .O17697 | 8.97 
24 .018235 | 8.98 
25 .018774 | 8.97 
26 .019312 | 8.97 
27 .019850 | 8.95 
28 -020887 | §.95 
29 . 020924 8.95 
30 .021461 8.98 
31 | 9.021997 | 8.93 
82 .022533 | 8.93 
33 .023069 | 8.92 
34 .023604 | 8.92 
35 .024189 | 8.90 
36 024673 8.92 
37 .025208 | 8.90 
38 .025742 | 8.88 
39 .026275 | 8.88 
40 | .026808 | 8.88 
41 | 9.027341 | 8.88 
42 027874 | 8.87 
43 .028406 |} 8.87 
44 .028938 | 8.85 
45 .029469 | 8.85 
46 .030000 | 8.85 

ff .030531 8.85 
48 .031062 | 8.83 
49 .031592 | 8.83 
50 .082122 |; 8.82 
51 | 9.082651 8.82 
52 .033180 | 8.82 
53 .033709 8.80 
54 .034237 | 8.80 
55 .034765.| 8.80 
56 .035293 | 8.78 
57 .035820 | 8.78 
58 .036347 | 8.78 
59 .036874 | 8.78 
60 | 9.037401 | 8.77 





Ex. sec. 


9.051546 
052155 
052763 
053371 
-053979 
-054586 
055193 
-055800 
-056406 
057012 

~ 057618 


9.058224 


063659 


9.064262 
064864 
-065466 
-066067 
066668 
067269 
067870 
068470 
.069070 
.069670 


9.070269 
.070868 
071467 
072065 
072663 
073261 
073859 
074456 
075053 
075649 


9.076246 
076842 
077487 
078033 
078628 
079223 
079817 
080412 
-081006 
081599 


9.082193 


“086929 
9.087520 





Dra" 





10.15 
10.18 
10.13 
10.138 
10.12 
10.12 
10.12 
10.10 
10.10 
10.10 
10.10 


10.08 
10.08 
10.07 
10.08 
10.07 
10.05 
10.07 
10.05 
10.03 
10.05 


10.03 
10.03 
10.02 
10.02 
10.02 
10.02 
10.00 
10.00 
10.00 

9.98 


9.98 
9.98 


WHOWOOOOOOowe sd 
(oe BO oe od he ef e oe oho oho oe 2) 
Go OT OT OT $2 2 2 D230 














= 
SUWMONIHAM FP WMH OS 


9.037401 
-037927 
. 038452 
088978 
039503 
-040027 
040552 
041076 


-047858 


9.048377 
.048896 
049415 
-049933 
050451 
050969 
051487 
052004 
052520 
053037 


9.053553 
054069 
054584 
055099 
.055614 
056129 
056643 
057157 
-057670 
058183 


9.058696 
-059209 
059721 
.060233 
060745 
061256 
-061767 
062277 
062788 
063298 


9.063807 
.064317 
064826 
-065335 
065848 
.066351 
.066859 
067366 
067874 

9.068380 











27° 


Dot. 


Ex. sec. 


8.77 | 9.087520 


8.75 


8.67 


Coco OT OLOTOt=2 3 DAO OS SSSSSSBBRR orn 


RRR RS hor Loon 


BIWAWNVINVNDDHOS DOwoww 


ee ip. 
or 


.088110 
088700 
089290 
.089880 
-090469 
091058 
091647 
092235 
092823 
093411 


9.093998 
-094586 
.095173 
095759 
.096346 
096932 
.097518 
-098103 
.098689 
099274 


9.099858 

- 100448 
- 101027 
101611 
102194 
10277 
- 103361 
. 103943 
. 104526 
. 105108 


9.105690 
106271 
- 106853 
107434 
- 108015 
- 108595 
-109175 
- 109755 
110835 
110914 


9.111494 
112072 
112651 
118229 
-113807 
114885 
-114963 
- 115540 
116117 
116693 


9.117270 











Dies 


| 


IID AADWDDMDOMDODOOOD 


eer! 
IIR 


g e 


DAD DP PDAUARUARWADRWRWD AWAD 
SOO WOW WWWWWOCD Ot 


CVrOT Or or or 





AND EXTERNAL SECANTS 149 

































































28° 29° 

, Vers. | D. 1’: | Ex. see. | D. 1’. , Vers. | D.1".| Ex. sec. |D. 1’. 
0 | 9.068380 | 8.45 | 9.122445 | 9.57 0 | 9.098229 | 8.15 | 9.156410 | 9.30 

1 .068887 | 8.43 .123019 | 9.57 1 .098718 | 8.13 .156968 | 9.32 
2 .069393 | 8.438 .128598 | -9.55 2 .099206 | 8.12 .157527 | 9.28 
3 .069899 | 8.43 .124166 | 9.55 3 .099693 | 8.13 .158084 | 9.30 
4 .070405 ; 8.42 .124739 | 9.53 4 .100181 | 8.12 .158642 | 9.30 
5 .070910 | 8.42 wi2ba11 |) 9.55 5 .100668 | 8.12 .159200 | 9.28 
6 .071415 8.40 . 125884 9.53 6 .101155 | 8.12 .159757 9.28 
4 .071919 8.42 . 126456 9.53 VF .101642 | 8.10 . 1603814 9.27 
8 .072424 | 8.40 .127028 | 9.52 8 .102128 | 8.10 .160870 | 9.28 
9 .072928 | 8.40 .127599 | 9.538 9 .102614 | 8.10 .161427 | 9.27 
10 .073432 | 8.38 .128171 9.52 || 10 .103100 | 8.08 .161983 | 9.27 
11 | 9.073935 | 8.38 | 9.128742 | 9.52 |] 11 | 9.108585 | 8.08 | 9.162539 | 9.27 
12 074438 | 8.38 .129313 | 9.50 || 12 .104070 | 8.08 -168095 | 9.25 
13 074941 | 8.37 .129883 | 9.50 || 138 .104555 | 8.08 .163650 | 9.25 
14 075443}. 8.38 .1380453 | 9.50 || 14 .105040 | 8.07 .164205 | 9.25 
15 075946 | 8.385 .1381023 | 9.50 || 15 .105524 | 8.07 .164760 | 9.25 
16 076447 | 8.387 .131593 | 9.50 || 16 .106008 | 8.05 .165315 | 9.25 
17 076949 | 8.35 .1382163 | 9.48 ti .106491 | 8.07 .165870 | 9.238 
18 .077450 | 8.35 .1382782 | 9.48.1) 18 .106975 | 8.05 .166424 | 9.23 
19 077951 8.35 . 133301 9.48 19 .107458 | 8.05 .166978 9.23 
20 078452 | 8.33 .1383870 | 9.47 || 20 .107941 | 8.03 .167582 | 9.22 
21 | 9.078952 | 8.33} 9.134438 | 9.47 || 21 | 9.108423 | 8.05 | 9.168085 | 9.23 
22 .079452 | 8.33 .135006 | 9.47 || 22 .108906 | 8.03 .168639 | 9.22 
23 .079952 | 8.82 .185574 | 9.47 || 23 .109388 | 8.02 .169192 | 9.22 
24 .080451 | 8.32 .186142 | 9.45 || 24 .109869 | 8.03 .169745 | 9.20 
25 .080950 | 8.32 .136709 | 9.47 || 25 .110851 | 8.02 .170297 | 9.22 
26 .081449 | 8.32 .187277 | 9.45 |} 26 .110832 | 8.02 .170850 | 9.20 
PH .081948 ; 8.30 .187844 | 9.43 || 27 -1113138 | 8.00 .171402 | 9.20 
28 .082446 | 8.30 .138410 | 9.45 || 28 .111793 | 8.00 .171954 | 9.18 
29 .082944 | 8.28 .188977 | 9.43 || 29 .112273 | 8.00 .172505 | 9.20 
30} .083441 | 8.30 | .139543 | 9.43 || 30 .112753 | 8.00 .178057 | 9.18 
31 | 9.083939 | 8.28 | 9.140109 | 9.42 || 31 | 9.113233 | 8.00 | 9.173608 | 9.18 
32 .084486 | 8.27 .140674 | 9.43 || 32 .118713 | 7.98 .174159 | 9.18 
33 .0849382 | 8.28 .141240 | 9.42 || 33 .114192 | 7.98 .174710 | 9.17 
34 .085429 | 8.27 .141805 | 9.42 || 34 .114671 | 7.97 .175260 | 9.17 
35 .085925 | 8.25 . 142370 | 9.40 || 35 .115149 | 7.97 .175810 | 9.17 
36 .086420 | 8.27 .1429384 | 9.42 || 36 .115627 | 7.97 .176360 | 9.17 
37 .086916 | 8.25 .143499 | 9.40 || 37 .116105 | 7.97 .176910 | 9.17 
38 .087411 | 8.25 .144063 | 9.40 || 38 .116583 | 7.97 .177460 | 9.15 
39 .087906 | 8.23 .144627 | 9.38 || 39 .11'7061 | 7.95 .178009 | 9.15 
40 .088400 | 8.25 .145190 | 9.40 || 40 .117538 | 7.95 .178558 | 9.15 
41 | 9.088895 | 8.23 | 9.145754 | 9.38 || 41 | 9.118015 ; 7.93 | 9.179107 | 9.15 
42 .089389: | 8.22 .146317 | 9.38 || 42 .118491 | 7.95 .179656 | 9.13 
43 .089882 | 8.23 .146880 | 9.87 || 43 .118968 | 7.93 .180204 | 9.13 
44 .090376 | 8.22 .147442 | 9.38 || 44 .119444 | 7.92 .180752 | 9.13 
45 .090869 | 8.22 .148005 | 9.387 || 45 .119919 | 7.90 .181300 | 9.13 
46 .091362 | 8.20 .148567 | 9.37 || 46 .120395 | 7.92 .181848 | 9.12 
47 .091854 | 8.20 .149129 | 9.35 || 47 .120870 | 7.92 .182395 | 9.13 
48 .092346 | 8.20 .149690 | 9.35 || 48 .121845 | 7.92 .182943 | 9.12 
49 .092838 | 8.20 150251 9.37 || 49 .121820 | 7.90 .188490 | 9.10 
50 .093330 ; 8.15 .150813 |; 9.33 || 50 . 122294 | 7.90 .184036 | 9.12 
51 | 9.093821 | 8.18 | 9.151373 | 9.35.|| 51 | 9.122768 | 7.90 | 9.184583 | 9.10 
52 .094312 | 8.18 .1519384 | 9.33 ||. 52 123242 | 7.88 .185129 | 9.10 
53 094803 | 8.17 .152494 | 9.35 || 53 128715 | 7.90 .185675 | 9.10 
54 -095293.| 8.17 |°°.1538055 | 9.32 || 54 .124189 | 7.88 . 186221 9.10 
55 .095783 | 8:17 .153614 | 9.383 || 55 .124662 | 7.87 .186767 | 9.08 
56 .096273 | 8.17 .154174 | 9.82 |) 56 .125134 | 7.88 .187312 | 9.10 
57 .096763 | 8.15 .1547383 | 9.33 || 57 .125607 | 7.87 .187858 | 9.08 
58 .097252 | 8.15 .155293 | 9.380 || 58 .126079 | 7.87 .188403 | 9.07 
59 097741} 8.13 155851 9.382 || 59 .126551 | 7.85 .188947 | 9.08 
GO | 9.098229 | 8.15 | 9.156410 | 9.30 || 60 | 9.127022 | 7.87 | 9.189492 | 9.07 
ee 


150 TABLE XXVI.—LOGARITHMIC VERSED SINES 








, Vers. 
0 | 9.127022 
1 .127494 
2 . 127965 
a - 128436 
4 . 128906 
5 . 129376 
6 . 129846 
q . 180316 
8 .130785 
9 .181255 
10 131724 
11 | 9.182192 
12 .1382660 
13 . 188129 
14 . 138596 
15 . 134064 
16 .184531 
17 . 134998 
18 . 135465 
19 .135931 
20 . 186397 
21 | 9.136863 
Q2 -187329 
23 .1387794 
24 . 138260 
25 .1388724 
26 . 139189 
27 . 139653 
28 .140117 
29 .140581 
30 . 141045 
31 | 9.141508 
32 .141971 
33 . 142484 
34 . 142896 
85 . 148358 
36 . 143820 
Sy . 144282 
38 . 144743 
39 . 145204 
40 . 145665 
41 | 9.146126 
42 . 146586 
43°} .147046 
44 . 147506 
45 .147966 
46 . 148425 
47 . 148884 
48 .149343 
49 . 149801 
50 . 150259 
51 | 9.150717 
52 .151175 
53 . 151633 
54 . 152090 
55 - 152547 
56 . 1538003 
57 . 153460 
58 153916 
59 . 154372 
60 | 9.154828 











= 
fer) 
Ps; 





Bo ae ee eet 
OTS? D2 D2 DI. SPN. S2> Sd lorKor) for) 





Ex. sec. 


190036 
. 190580 
191124 
.191668 
192211 
192754 
198297 
193840 
194382 
194925 


9.195467 
196009 
. 196550 
197092 
197633 
198174 
198715 
199255 
199795 
200335 


9.200875 
201415 
201954 
202494 
208032 
.203571 
-204110 
204648 
205186 
205724 


9.206262 
206799 
207337 
207874 
208410 
. 208947 
. 209483 
210020 
210556 
211091 


9.211627 
212162 
212697 
218282 
213767 
214301 
214836 
215370 
215904 
216437 


9.216971 
217504 
218037 
218570 
219102 
219635 
220167 
220699 

221231 


9.221762 





9.189492 


9.02 


WWMM WNW WOWZIO VQBRAVNVDBHVWSAHS SS38 


WOMMOOSOoowows wo co 3 sos © Vee jeile fief ye) oo 
ESSSEE os sos oo > 


GO GO GO G0 GD GD GO OD 


Qo oO 
OUTOTE 2-22 D3 DD 
































31° 
Vers. |D. 1°.) Ex. set. |D. 1”: 
9.154828 | 7.58 | 9.221762 8.85 
.155283 | 7.58 . 2222938 8.87 
.155788 | 7.58 222825 8.83 
.156193 | 7.58 . 223855 8.85 
.156648 | 7.57 . 223886 8.85 
.157102 | 7.57 224417 8.83 
.157556 | 7.57 224947 8.83 
.158010 | 7.57 225477 8.83 
. 158464 | 7.55 226007 8.83 
.158917 | 7.55 2265387 8.82 
.159870 | 7.55 . 227066 8.82 
9.159823 | 7.55 | 9.227595 8.838 
. 160276 | 7.53 , 228125 8.80 
.160728 | 7.53 . 228653 8.82 
.161180 | 7.53 ,229182 8.82 
.161682 | 7.52 ,229711 8.80 
. 162088 | 7.53 . 280289 8.80 
162080" |) 7 coe 280767 8.80 
.162986 | 7.52 .231295 8.78 
168487 | 7.50 ,231822 8.80 
.163887 | 7.52 . 282850 8.78 
9.164338 | 7.50 | 9.232877 8.78 
.164788 | 7.48 . 238404 8.78 
165287! | 7.50 . 233931 8.78 
.165687 | 7.48 .234458 8.7 
.166136 | 7.48 . 234984 8.77 
.166585 | 7.48 . 235510 8.77 
.167034 | 7.48 . 236036 8.7 
.167483 | 7.47 . 236562 8.77 
.167931 | 7.47 . 237088 BF 
.168379 | 7.47 . 237613 8.77 
9.168827 | 7.47 | 9.238139 8.7 
.169275 | 7.45 , 238664 8.7 
.169722 | 7.45 . 239189 8.73 
.170169 | 7.45 239713 8.75 
.170616 | 7.43 . 240238 8.73 
.171062 | 7.45 . 240762 8.73 
.171509 | 7.43 .241286 | 8.73 
.171955 | 7.42 . 241810 8.7 
.172400 | 7.43 . 242833 8.7% 
.172846 | 7.42 .242857 | 8.72 
9.178291 | 7.42 | 9.248380 Sor 
173786 | %.42 . 248903 8.72 
.174181 | 7.42 . 244426 8.72 
.174626 | 7.40 . 244949 8.70 
-175070 | 7.40 . 245471 8.72 
.175514 | 7.40 . 245994 8.70 
.175958 | 7.40 . 246516 8.7 
.176402 | 7.38 . 247038 8.68 
.176845 | 7.38 . 247559 8.70 
177288 | 7.88 | .248081 | 8.68 
9.177731 | 7.88 | 9.248602 8.68 
178174 |-7.37 . 2491238 8.68 
-178616 | 7.387 . 249644 8.68 
.179058 | 7.37. . 250165 8.68 
.179500 | 7.387 . 250686 8.67 
.179942 | 7.85 +251206 8.67 
. 180383 | 7.37 . 251726 8.67 
.180825 | 7.38 . 252246 8.67 
.181265 | 7.385 . 252766 8.67 
9.181706 | 7.85 | 9.258286 8.65 


AND EXTERNAL SECANTS 151 






















































































32° 33° 

4 Vers. | D.1".| Ex. sec. | D, 1’. f Vers. |D.1".) Ex. sec. | D. 1’. 

0 | 9.181706 7°35 9. 253286 8.65 0 | 9.207714 | 7.10 | 9.284122 8.48 

1 .182147 7.33 . 2538805 8.65 1 .208140 | 7.10 . 284631 8.47 

2 .182587 7.33 , 2543824 8.65 2 .208566 | 7.10 .285139 8.47 

3 .183027 “32 . 254843 8.65 8 . 208992 | 7.10 285647 8.47 

4 .183466 7.33 . 255862 8.65 4 .209418 | 7.08 . 286155 8.47 

5 183906 7.32 . 255881 8.63 5 .2098438 | 7.08 . 286663 8.45 

6 184845 7 32 . 256399 8.65 6 .210268 | 7.08 . 287170 8.47 

7 184784 7.32 . 256918 §.63 q .210693 | 7.08 .287678 8.45 

8 185223 ? 32 257436 8.63 8 .211118 | 7.08 . 288185 8.45 

9 185662 7.80 . 257954 8.62 ||. 9 .2115438 | 7.07 . 288692 8.45 
10 186100 7 30 258471 8.63 10 .211967 | 7.07 . 289199 8 438 
11 | 9.186538 7.30 | 9.258989 8.62 || 11 | 9.212891 | 7.07 | 9.289705 8.45 
12 .186976 7.28 .259506 8.62 12 .212815 | 7.07 . 290212 8.43 
13 .187413 7.30 .260028 8.62 13 . 2138239 | 7.05 . 290718 8.43 
14 187851, 7 28 . 260540 8.62 || 14 .2138662 | 7.05 . 291224 8.43 
15 . 188288 TRV . 261057 8.62 || 15 .214085 | 7.05 .291730 8.43 
16 . 188724 7.28 .261574 8.60 16 .214508 | 7.05 . 292236 8.43 
17 .189161 7.27 .262090 8.60 Y; .2149381 | 7.05 .292742 8.42 
18 . 189597 7.28 .262606 8.60 18 .215854 | 7.08 .298247 8.48 
19 . 190034 7.25 .263122 8.60 19 PISTG. T.08 . 2937538 8.42 
20 . 190469 Ti 20 . 263638 8.60 || 20 .216198 | 7.08 . 294258 8.42 
21 | 9.190905 7.27 | 9.264154 8.58 || 21 | 9.216620 | 7.038 | 9.294763 8.42 
22 191841 | 7.25 . 264669 8.58 || 22 .217042 | 7.02 . 295268 8.40 
23 .191776 7.20 265184 8.60 || 23 .217468 | 7.02 .295772 8.42 
24 .192211 7 28 265700 8.57) 1) 24 .217884 | 7.02 .296277 8.40 
25 .192645 5) 266214 8.58 || 25 -,218805 | 7.02 . 296781 8.40 
26 . 193080 7 23 266729 8.58 || 26 .218726 | 7.00 .297285 8.40 
27 . 193514 7.23 267244 8.57 || 27 .219146 | 7.02 .297'789 8.40 
28 .1938948 7.23 267758 8.57 || 28 .219567 | 7.00 . 298293 8.40 
29 . 194382 7.22 .268272 8.57 || 29 .219987 | 7.00 . 298797 8 388 
20: .194815 7.23 1 .268786 8.57 || 30 .220407 | 6.98 . 299800 8.388 
81 | 9.195249 7.22 | 9.269300 8.57 || 31 | 9.220826 | 7.00 | 9.299803 8.40 
32 . 195682 7 22 .269814 8.55 || 32 .221246 | 6.98 .800807 8.37 
33 .196115 7.20 . 270827 8.55 || 33 .221665 | 6.98 . 800809 8.38 
34 . 196547 7.22 .270840 8.57 || 34 . 222084 | 6.98 .801812 8.38 
35 . 196986 7.20 .271354 8.53 || 85 . 222508 | 6.97 301815 8.37 
36 .197412 7.20 . 271866 8.55 || 36 . 222921 | 6.98 3023817 8.38 

v4 .197844 7.18 . 2723879 8.65: ||: 37 223840 | 6.97 302820 8.387 
38 .198275 7.20 272892 8.53 || 38 .223758 | 6.97 808822 8.37 
39 .198707 7.18 .2738404 8.53 || 39 . 224176 | 6.95 803824 8.35 
40 .199138 7.18 . 273916 8.53 || 40 .224598 | 6.97 304825 8.37 
41 | 9.199569 7.18 | 9.274428 8.53 41 | 9.225011 | 6.95 | 9.304827 8.35 
42 . 200060 7 ALL . 274940 8.53 42 225428 | 6.95 . 805328 8.37 
43 . 200430 7.18 .275452 8.52 43 .225845 | 6.95 . 805880 8.35 
44 . 200861 WAG. . 275963 8.52 44 . 226262 | 6.93 . 3063831 8.35 
45 . 201291 7.15 276474 8.538 || 45 .226678 | 6.95 . 3806832 8.35 
46 . 201720 mei g .276986 8.50 || 46 .227095 | 6.93 .807383 8.338 
47 . 202150 7.415 277496 8.52 || 47 227511 | 6.98 .807883 8.35 
48 20257 “di . 278007 8.52 || 48 . 227927 | 6.92 . 808834 8.33 
49 . 203008 Y 15 .278518 8.50 || 49 .228342 | 6.93 3808834 8.38 
50 , 203437 see 3 . 279028 8.50 || 50 .228758 | 6.92 .3809834 8.33 
51 | 9.208866 7.13 | 9.279588 8.50 || 51 | 9.229173 | 6.92 | 9.809834 8.33 
52 . 204294 7 16 . 280048 8.50 52 . 229588 | 6.92 .3810384 8.33 
53 . 204723 7 18 .280558 8.50 || 53 .230003 | 6.92 . 810834 8.32 
54 . 205151 7.12") .281068.| 8.48 || 54 .230418 | 6.90 .3113833 8.32 
55 .205578 fai 3 .281577 8.50 || 55 . 2308382 | 6.90 .811882 8.32 
56 . 206006 7.13 . 282087 8.48 || 56 .231246 | 6.90 312331 8.32 
57 . 206433 VA 282596 8.48 || 57 . 231660. | 6.90 312830 8.32 
58 206860 7 AR .283105 8.48 || 58 232074 | 6.88 8183829 8.32 
59 .207287 | 7.12 283614 8.47 || 59 . 232487 | 6.90 .313828 8.30 
60 | 9.207714 7.10 | 9.284122 8.48 60 | 9.282901 6.88 | 9.314326 8.32 








152 TABLE XXVI.—LOGARITHMIC VERSED SINES 


















































34° 35° 
‘ Vers. Dele Halse seca aD eal / Vers. |D.1".| Ex. sec. |D. 1’. 
0 | 9.232901 6.88 | 9.314826 8.32 0 | 9.257814 | 6.67 | 9.343949 8.15 
1 .233314 6.88 .814825 8.30 1 .200714 | 6.68 .344438 8.15 
2 PSS YPA 6.87 .oldd2d 8.30 Q .258115 | 6.67 .3844927 8.15 
3 .234139 6.88 015821 8.30 3 .208015 | 6.67 . 845416 8.138 
4 234552 6.87 .3163819 8.30 4 .258915 | 6.65 .3845904. 8.15 
5 . 234964 6.87 .3816817 8.28 5 .259314 | 6.67 .846393 8.13 
6 . 230376 6.87 .317314 8.28 6 .209714 | 6.65 .846881 | . 8.138 
@ .235788 6.85 .3817811 8.30 q .260113 | 6.65 .847369 8.13 
8 .236199 | 6.87 .3818309 8.28 8 .260512 | 6.65 . 847857 8.138 
9 . 236611 6.85 .318806 8.28 9 -260911 | 6.65 .848345 8.138 
10 . 237022 6.85 .8193803°| 8.27 10 -261810 | 6.65 .848833 8.13 
11 | 9.237483 6.85 | 9.319799 8.28 |} 11 | 9.261709 | 6.63 | 9.349821 8.12 
12 237844 6.83 .320296 8.27 12 .262107 | 6.63 .349808 8.12 
13 . 238254 6.85 .820792 8.28 || 18 . 262505 | 6.63 .800295 | 8.12 
14 . 238665 6.83 .821289 8.27 || 14 .262903 | 6.63 . 80782 8.12 
15 . 239075 6.83 .821785 Sak 15 .263301 | 6.62 .3851269 8.12 
16 . 239485 6.82 .3822281 8.25 16 .263698 | 6.63 .3801756 8.12 
17 239894 6.83 .3822776 8.27 17 .264096 | 6.62 .3022438 8.12 
18 . 240304 6.82 .820272 8.27 18 . 264493 | 6.62 .3802730 8.10 
19 .2407138 6.82 .823768 8.25 19 -264890 | 6 62 .3808216 8.10 
20 241122 6.82 .3824263 8.25 || 20 . 265287 | 6.60 .308102 8.10 
21 | 9.241531 6.82 | 9.824758 8.25 || 21 | 9.265683 | 6.62 | 9.354188 8.10 
22 | - 241940 6.82 .820253 8225 Vege .266080 | 6.60 .804674 8.10 
2B} . 249348 6.80 .820748 8.25 || 23 .266476 | 6.60 .805160 8.10 
24 242756 6.80 .3826243 8.23 || 24 .266872 | 6.58 . 300646 8.08 
25 .243164 6.80 .326737 8.25 || 25 -267267 | 6.60 .306131 8.10 
26 248572 6.80 827232 8.23 || 26 -267663 | 6.58 .806617 8.08 
27 . 243980 6.78 .3827026 S23 qNe7 .268058 | 6.58 .3857102 8.08 
28 244387 6.78 .328220 8.23 || 28 .268453 | 6.58 .B07D87 8.08 
29 244794 | 6.7 828714 | 8.22 || 29 .268848 | 6.58 .858072 | 8.08 
30 .245201 | 6.78 .3829207 | 8.23 || 80; .269248 | 6.58 .358557 | 8.08 
81 | 9.245608 | 6.77 | 9.829701 8.23 || 31 | 9.269638 | 6.57 | 9.359042 | 8.07 
32 .246014 6.78 .3830195 8.522 ||| 32 .270032 | 6.57 .309526 8.08 
33 246421 6.77 .3380688 8.22 || 33 .270426 | 6.57 3860011 8.07 
34 . 246827 6.77 .831181 8.22 || 34 . 270820 | 6.57 360495 8.07 
35 247233 6.77 .3831674 8.22 || 35 -271214 | 6.57 360979 8.07 
36 . 247639 6.75 .832167 8.20 || 36 .271608 | 6.55 361463 8.07 
or . 248044 6.75 .832659 Sree wee? .272001 | 6.55 361947 8.07 
38 . 248449 6.75 .3833152 8.20. || 38 . 272394 | 6.55 .3862431 8.05 
89 4 .248854 6.75 .333644 8.22 || 89 272787 | 6.55 .3862914 8.07 
40 . 249259 6.75 .834137 8.20 || 40 . 278180 | 6.53 . 868398 8.05 
41 | 9.249664 | 6.73 | 9.334629 | 8.20 || 41 | 9.278572 | 6.55 | 9.363881 | 8.05 
42 . 250068 6.75 .830121 8.18 42 . 273965 | 6.53 . 364364 8.05 
43 .250473 6.73 .38380612 8.20 || 43 .274357 | 6.53 .3864847 8.05 
44 . 200877 6.73 .336104 8.18 || 44 .274749 | 6.53 .365380 8.05 
45 .201281 6.72 . 3836595 8.20 || 45 .275141 | 6.52 .3865813 8.03 
46 . 251684 6.73 .837087 8.18 || 46 .275582 | 6.53 . 366295 8.05 
47 . 252088 6.72 .887578 8.18 || 47 . 275924 | 6.52 .3866778 8.03 
48 . 252491 6.72 . 3838069 8.18 || 48 .276815 | 6.52 . 367260 8.03 
49 252894 6.72 . 838560 8.17 49 276706 | 6.52 . 867742 8.03 
50 1258297) 6.70 .3839050 | 8.18 || 50 277097 | 6.52 .368224 | 8.03 
51 | 9.253699 6.72 | 9.339541 8.17 || 51 | 9.277488 | 6.50 | 9.368706 8.03 
652 . 254102 6.70 . 3840031 8.18 || 52 277878 | 6.50 .3869188 8.03 
53 .204504 6.70 .3840522 8.17 || 538 .278268 | 6.50 . 3869670 8.02 
54 .254906 6.70 .341012 8.17 || 54 .278658 | 6.50 .ov0151 8.02 
55 . 255308 6.68 .3841502 Sato) top .279048 | 6.50 370632 8 03 
56 . 205709 6.70 .3841991 8.17 || 56 . 279438 | 6.48 .3871114 8.02 
57 .256111 6.68 342481 Sela ioe . 279827 | 6.50 .311595 8.02 
58 . 296512 6.68 .3842971 8.15 || 58 .280217 | 6.48 .872076 8.00 
59 . 256913 6.68 . 348460 8.15 || 59 .280606 | 6.48 .872556 8.02 
60 | 9.257314 6:67 | 9.343949 8.15 |! 60 | 9.280995 | 6.47 | 9.373037 8.02 - 











AND EXTERNAL SECANTS 153 
































36° 37° 
| 

Vers..-|D. 1". |Ex.-see. | D.1".|| ” Vers. Dri? Wisbxasees Desi". 

0 | 9.280995 6.47 | 9.373037 02 0 | 9.803983 | 6.28 | 9.401634 7.88 
4 .281383 6.48 .378018 00 1 .3804360 | 6.30 .402107 7.88 
2 .281772 6.47 .373998 00 2 .3804738 | 6.28 .402580 7.87 
3 . 282160 6.47 874478 00 3 .805115 | 6.28 .408052 7.87 
4 .289548 |. 6.47 .374958 00 4 .3805492 | 6.27 . 403524 7.88 
3) . 282936 6.47 .3875488 00 5 .3805868 | 6-28 .403997 7.88 
6 . 283824 6.47 .3815918 00 6 .306245 | 6.27 .404469 7 87 
ve . 283712 6.45 .816398 98 7 .3806621 | 6.28 .404941 (ear 
8 . 284099 6.45 3876877 00 8 .3806998 | 6.27 .405412 Toate 
9 . 284486 6.45 18T735¢ 98 || 9 .807874 | 6.25 . 405884 7 87 
10 . 284873 6.45 .3807836 98 |} 10 .807749 | 6.27 .406356 7.85 
11 | 9.285260 6.45 | 9.378315 98 |} 11 | 9.308125 | 6.27 | 9.406827 7.85 
12 . 285647 6.438 .378794 98 12 .808501 | 6.25 .407298 Phev 
3 . 286033 6.438 .3819273 98 13 .308876 | 6.25 407770 4.85 
14 . 286419 6.43 879752 98 || 14 .3809251 | 6.25 .408241 7.85 
15 . 286805 6.43 .880231 97 15 - 3809626 | 6.25 .408712 7.85 
16 . 287191 6.438 .880709 98 16 .810001 | 6.23 .409183 7.83 
17 287577 6.42 .881188 97 if .3810875 | 6.25 .409653 7.85 
18 .287962 6.40 .3881666 97 18 .310750 | 6.23 .410124 7.83 
19 .288348 6.42 ,38021 44 97 19 -311124 | 6.23 .410594 7.85 
290 . 2887383 6.42 . 802622 97 || 20 -311498 | 6.23 .411065 7.83 
21 | 9.289118 6.40 | 9.388100 95 || 21 | 9.311872 | 6.22 | 9.411535 7.83 
99 . 289502 6.42 .888577 97 || 22 .912245 | 6.23 .412005 7.83 
93 .289887 | 6.40 . 884055 95 |+ 23 .312619 | 6.22 .412475 7.83 
94 . 29027 6.40 .384532 97 || 24 .3812992 | 6.22 .412945 7.83 
95 . 290655 6.40 | 385010 95 |) 25 .313365 | 6.22 .413415 7.82 
6 . 291039 6.40 .3885487 95 || 26 .3138738 | 6.2 .413884 (283 
ye . 291423 6.40 .885964 .95 Dae .014111 | 6.22 .414354 7.82 
98 . 291807 6.38 .3886441 .95 |} 28 .314484 | 6.20 .414823 7.83 





29 314856 6.20 .415293:'] 7.82 
80 | .3815228 | 6.20 415762 | 7.82 


31 | 9.315600 | 6.20 | 9.416231 | 7.82 
82 | .315972 | 6.20 | .416700 | 7.80 


29 | .292190 | 6.38 | .386918 
30 | .292573 | 6.38 |. .3873894 


31 | 9.292956 | 6.38 | 9.387871 
32 | .2933389 | 6.88 | .388347 





WOWOOOOH Ww 
WWW COCEWOICO Orcs 
oo 
co 


























RRANRA NNR NIRA DENNER SESENENE NEN 2 2 -D~ID.E.DMHDH O00 


~I72 
@0 00 
ao Go 
ror 
S 


33 | .2938722 | 6.87 | .388824 f .3816344 | 6.20 | .417168 | 7.82 
34 | .294104 | 6.37 . 3889300 34 | .316716 | 6.18 | .417637 | 7.82 
35 | .294486 | 6.37 889776 385 817087 | 6.18 | .418106 | 7.80 
36 | .294868 | 6.37 | .390252 386 | .317458 | 6.18 | .418574 | 7.80 
87 | .295250 | 6.37 890727 7 | .817829 | 6.18 | .419042 | 7.82 
88 | ..295682 | 6.37 | .3891203 92 || 38 | .818200 | 6.18 | .419511 | 7.80 
39 | .296014 | 6.385 | .391678 93 || 89 | .3818571 | 6.17 | .419979 | 7.80 
40 | .296895 | 6.85 | .892154 92 || 40 | .818941 | 6.17 | .420447 | 7.80 
41 | 9.29677 6.35 | 9.392629 92 || 41 | 9.319311 | 6.18 | 9.420915 | 7.7 
42 | .297157 | 6.35 | .893104 92 || 42 | .819682 | 6.15 .421382 | 7.80 
43 | .297538 | 6.33 | .393579 92 || 43 820051 | 6.17 | .421850 | 7.78 
44) .297918 | 6.35 894054 92 |) 44 | .820421 | 6.17 | .422317 | 7.80 
45 | .298299 | 6.33 | .394529 90 || 45 820791 | 6.15 422785 | 7.78 
46 | .298679 | 6.33 | .395003 92 || 46 .821160 | 6.17 423252 | 7.78 
71 .299059 | 6.33 | .395478 90 7 021530 | 6.15 423719 | 7.7 
48 | .299489 | 6.383 3895952 90 || 48 .821899 | 6.13 | .424186 | 7.78 
49 | .299819 | 6.32 3896426 90 |) 49 | .822267 | 6.15 | .424653 | 7.78 
50 | .300198'| 6.32 .396900 90 |; 50 | .322636 | 6.15 .425120 | 7.78 
51 | 9°300577 | 6.33 | 9.397374 90 |; 51 | 9.323005 | 6.13 | 9.425587 | 7:7 
52 | .800957 | 6.30 3897848 90 || 52 823373 | 6.13 .426053 | 7.78 
53 | .301335 | 6.32 | .398322 88 || 53 | .323741 |. 6.13 .426520 | 7.77 
54 | -.301714 | 6.32%s .398795 90 || 54 | .824109 | 6.13 .426986 | 7.77 
55 | .302093 | 6.30 899269 88 || 55 |  .824477 | 6.13 427452 | 7 
56 | .302471 | 6.30] .399742 88 || 56]  .824845 | 6.12 427918 | 7.77 
57 | .302849 | 6.30} .400215 88 || 57 825212 | 6.18 428884 | 7.77 
58 | .303227 | 6.30 | .400688 88 || 58 | .325580 | 6.12 | .428850 | 7.77 
59 303605 | 6.30 | .401161 59 825947 | 6.12 429316 | 7.77 








60 | 9.303983 | 6.28 | 9.401634 9.326314 | 6.12 | 9.429782 | 7.75 
a eee 


154 TABLE XXVI.—LOGARITHMIC VERSED SINES 













































































FHFRSHAAAS SABINA BSS SRSSSSSSSS SSSSRS 











38° 39° 
‘ Vers. »)D. 1. y Ex. seca) Do 1. 4 | Vers. |D. 1‘. | lax. seer |Di ¥. 
O | 9.326314 | 6.12 | 9.429782 | 7.7 0 | 9.348021 | 5.93 | 9.457518 | 7.65 
1} .826681 | 6.10 | .4380247 | 7.7 1 | .848877 | 5.95 | .457977 | 7.65 
2| .827047 | 6.12} .480713 | 7.75 2| .3487384 | 5.93 | .458486 | 7.65 
3 | .827414' 6.10 | .481178 | 7.75 || 3, .349090 | 5.93 | .458895 | 7.63 
4 827780 | 6.10 | .431643 | 7.75 4| .849446 | 5.93 | .459353 | 7.65 
5 | .828146 | 6.10) .482108 | 7.75 5 | .349802 | 5.93 | .459812 | 7.63 
6 | .3828512| 6.10} .482573 | 7.7 6 | .850158 | 5.93 | .460270 | 7.65 
7 | .828878 | 6.08} .433038 | 7.7 7 | .850514 | 5.92 | .460729 | 7.68 
8 | .829243 | 6.10] .483503 | 7.73 8 | .850869 | 5.93 | .461187 | 7.63 
9 329609 | 6.08 | .433967 | 7.75 9 | .3851225 | 5.92 | .461645 | 7.63 
10 829974 | 6.08 | .484432 | 7.73 || 10 | .851580 | 5.92 | .462108 | 7.63 
11 | 9.330339 | 6.08 | 9.434896 | 7.75 || 11 | 9.351935 | 5.92 | 9.462561 | 7.63 
12 | .330704 | 6.08 | 485361 | 7.73 || 12 | .852290 | 5.90 | .463019 | 7.68 
13} .831069 | 6.07 | .485825 | 7.73 || 18 | .352644 | 5.92] .463477 | 7.62 
14 | 831483 | 6.08 | .436289 | 7.73 || 14 | .852999 | 5.90} .463934 | 7.63 
15 | .331798 | 6.07 | .436753 | 7.7 15 | .353353 | 5.90 | .464392 | 7.62 
16 | .332162 | 6.07 | .487217 | 7.72 || 16 | .3858707 | 5.92 | .464849 | 7.63 
17 | .3382526 | 6.07 | .487680 | 7.73 || 17] .854062 | 5.88 | .465307 | 7.62 
18, .332890 | 6.07 | .488144 | 7.73 || 18 | .854415 | 5.90 | .465764 | 7.62 
19 | .333254 | 6.05 | .488608 | 7.72 || 19 | .354769 | 5.90 | .466221 | 7.62 
20 | .3883617 | 6.07 ; .489071 | 7.72 || 20 | .855128 | 5.88 | .466678 | 7.62 
21 | 9.338981 | 6.05 | 9.489534 | 7.72 || 21 | 9.355476 | 5.88 | 9.467185 | 7.62 
22 | .334844 | 6.05 | .489997 | 7.72 || 22 | .355829 | 5.88! .467592 | 7.62 
(23 | 334707 | 6.05 | .440460 | 7.72 || 23] .356182 | 5.88} .468049 | 7.62 
24 | .385070 | 6.03 | .440923 | 7.72 || 24] .856535 | 5.88 | .468506 | 7.60 
25 | 385432 | 6.05 | .441886 | 7.72 || 25; .856888 | 5.88 | .468962 | 7 
26 | .385795 | 6.03 | .441849 | 7.7 26 | .857241 | 5.87 | .469418 | 7 
2¢ | .886157 | 6.03 | .442312 | 7.70 || 27 | .857593 | 5.87 | .469875 | 7 
28 | .336519 | 6.03 | .44277 Ci 28} .3857945 | 5.87 | 470881 | 7. 
29 | .336881 | 6.038 | .448287 | 7.70 || 29) .858207 | 5.87 | 420787 | 7, 
30 | .3387248 | 6.03 | .448699 | 7.70 || 80 | .3858649 | 5.87 | .471243 | 7. 
31 | 9.337605 | 6.02 | 9.444161 | 7.70 || 81 | 9.359001 | 5.87 | 9.471699 | 7 
82 | .337966 | 6.03 | .444623 | 7.70 || 82 | .359853 | 5.85 | .472155 | 7 
33} .338328 | 6.02 | .445085 | 7.7 83 | .359704 | 5.87 | .472611 | 7 
34, .388689 | 6.02 | .445547 | 7.70 || 84) .860056 | 5.85] .473067 | 7. 
35 | .339050 | 6.02 | .446009 | 7.68 || 85 | .860407 | 5.85 | .473522 | 7. 
36 | .339411 | 6.00 | .446470 | 7.70 || 36 | .360758 | 5.88} .4738978 | 7. 
37 | = .8389771 | 6.02 | .446982 | 7.68 7 | .861108 | 5.85 | .474483 |) 7, 
38 | .340182 | 6.00 | .447393 | 7.70 || 88 | .361459 | 5.85 | .474888 | 7 
39 | .340492 | 6.00 | .447855 | 7.68 || 89 | .861810 | 5.83} .4753843 |) 7. 
40 340852 | 6.00 | .448316 | 7.68 || 40 | .3862160 | 5.83} .475798 | 7. 
41 | 9.841212 | 6.00 | 9.448777 | 7.68 || 41 | 9.862510 | 5.83 | 9.476253 | 7. 
42 | .341572 | 6.00! .449288 | 7.68 || 42 | .3862860/ 5.83} .476708 | 7. 
43; .841982 | 5.98 | .449099 | 7.68 || 43 | .368210 | 5.838) .477163 | 7. 
44 | .342291 | 6.00 | .450160 | 7.67 || 44) .3863560 | 5.82} .477618 | 7. 
45 | .342651 | 5.98 | .450620 | 7 68 || 45 | .363909 | 5.838 | .478072 |; 7. 
46 | .3848010 | 5.98 | .451081 | 7.67 || 46 | .364259 | 5.82 | .478527 | 7. 
47 | 348369 | 5.98 | .451541 | 7.68 || 47 | .864608 | 5.82); .478981 | 7. 
48 | .3438728 | 5.97 | .452002 | 7.67 || 48 | .864957 | 5.82 | .479435 | 7. 
49 | .3844086 | 5.98 | .452462 | 7.67 || 49} .365306 | 5.82 | .479890 | 7. 
50 | .844445 | 5.97 | .452022 | 7.67 || 50 | .865655 | 5.80 | .480844 | 7. 
51 | 9.344803 | 5.97 | 9.453382 | 7.67 || 51 | 9.366003 | 5.82 | ¥.480798 | 7. 
52; .345161 | 5.97 | .453842 | 7.67 || 52 | 3866352 | 5.80 | .481252) 7. 
53.| .3845519 | 5.97 | .454302 | 7.67 || 53 | .366700 | 5.80 481705 | 7. 
54 | 345877 | 5.97 | .454762 | 7.65 || 54 367048 | 5.80 482159 | 7. 
55 | .346235 | 5.95 | .455221} 7.67 || 55 | .367396 | 5.80 482613 | 7. 
56 | .346592 | 5.97 | .455681 | 7.65 || 56 | 867744 | 5.78 483066 | 7. 
57 346950 | 5.95 456140 | 7.67 || 57 368091 | 5.80 483520 | 7. 
58 | .347307 | 5.95 | .456600 | 7.65 || 58 368439 | 5.78 483973 | 7. 
59 | .347664 | 5.95 | .457059 | 7.65 |) 59 368786 | 5.78 | 484426 | 7. 
60 | 9.348021 | 5.93 1 9.457518 | 7.65 || 6U | 9.369183 | 5.78 ' 9 484879 | 7. 
—_—_—_—_—__eee_ern—_e_ee 


AND EXTERNAL SECANTS . 155 





40° 41° 





= 


4 Vers. | D. 1’. | Ex. sec. | D. ate Vers. | D.1".| Ex. sec. |D. 1’. 





















































0 | 9.369133 | 5.78 | 9.484879 | 7.55 || 0 | 9.389681 | 5.62 | 9.511901 | 7.45 
1] .369480 | 5.78 | 485332 | 7.55 |] 1} .890018 | 5.63 | .512848 | 7.47 
2] 369827 | 5.78] .485785 | 7.55 || 2] 1390356 | 5.63 | .512796 | 7.45 
3] .370174 | 5.77 | .486838 | 7.55 |/ 3] 1390604 | 5.62 | 518243 | 7.47 
4] .370520| §.78| 486691} 7.55 || 4] 391031 | 5.62 | 518601 | 7.45 
5! (370867 | 5.77} 487144 | 7.53 |] 5 | 1391368} 5.62 | 1514138 | 7.45 
6| (371213] 5.77 | .487596 | 7.55 || 6 | .391705 | 5.62] (514585 | 7.47 
7| (871559 | 5.77 | ..488049 | 7.53 || 7) .392042) 5.62 | °515083 | 7.45 
8| .371905 | 5.77! 1488501 | 7.53 || 8| 392379 | 5.62 | 515480 | 7.45 
9) .372251| 5.75 | .488053 | 7.55 || 9] 392716 | 5.60 | -bi5oe7 | 7.45 
10} .372596 | 5.77 |  .489406 | 7.53 || 10 | .898052 | 5.60 | 516874 | 7.43 
11 | 9.872942 | 5.75 | 9.489858 | 7.53 || 11 | 9.393388 | 5:60 | 9.516820 | 7.45 
12} .373287 | 5.75 | .490310 | 7.53 || 12 | .393724 | 5.62 | 517267 | 7.45 
18 | .373632 | 5.75; .490762 | 7.53 || 18 | 1394061 | 5.58 | .Bi7714 | 7.43 
14) 373977 | 5.75 | 1491214 | 7.52 || 14 | .394396 | 5.60 | 1518160 | 7.45 
15 | .374392-) 5.75 | .491665 | 7.53 || 15 | 1894732 | 5.60 | .5i8607 | 7.43 
16 | .374667 | 5.73 | 492117 | 7.53 || 16 | 1395068 | 5.58 | .519053 | 7.45 
17| 3875011 | 5.75 | 1492569 | 7.52 || 17 | .395403 | 5.58 | .519500 | 7.43 
18 | .375356 | 5.73 | .493020 | 7.52 || 18 | .895738 | 5.60 | .519946 | 7.43 
19} .375700 | 5.73 | 493471 | 7.58 || 19 | .896074 | 5.58 | -520392 | 7.43 
20| 876044 | 5.73 | 493083 | 7.52 || 20] .396409 | 5.57 | 1520838 | 7.43 
21 | 9.876388 | 5.73 | 9.494374 | 7.52 |] 21 | 9.996743 | 5.58 | 9.521284 | 7.43 
22 | 1876732 | 5.72] 494885 | 7.52 || 21] 1397078 | 5.53 | 1521730 | 7.43 
23] .377075'|- 5.73 | .495276 | 7.52 || 23] 307413 | 5.57 | 5eei76 | 7.42 
24| ‘a774ig | 5.72 | .495727 | 7.52 || 24] 397747 | 5.57 | .5e26e1 | 7.43 
25 | 377762 | 5.72| :496178 | 7.50 || 25 | -398081 | 5.57] 1593067 | 7.43 
26 | (878105 | 5.72| 496628 | 7.52 || 26) 398415 | 5.57 | 1588518 | 7.42 
Q7 | .378448 | 5.72 | .497079 | 7.52 |] 27 | .808749 | 5.57 | 1583958 | 7.43 
28} 1878791 | 5.70 | 497530 | 7.50 || 28) .399083 | 5.57 | 524404 | 7.42 
29 | 1879133 | 5.72| .497980| 7.52 || 29] 300417 | 5.55 | 524849 | 7.42 
30 | .379476 | 5.70| 1498430 | 7.50 || 30 | .399750 | 5.57 | c5e5204 | 7.42 
31 | 9.379818 | 5.72 | 9.498881 | 7.48 || 81 | 9.400084 | 5.55 | 9.525739 | 7.42 
32) (380161 { 5.70 | .499331 | 7.52 || 82] .400417 | 5.55 | 526184. | 7.42 
33 | 380503} 5.70 | .499781 | 7.50 || 33] .400750 | 5.55 | 526629 | 7.42 
34] 1380845} 5.68} :500231 | 7.50 || 84] .401083 | 5.55 | c5e7074 | 7.42 
85} 1381186 | 5.70| .500681 | 7.50 || 85 | .401416 | 5.53 | ‘527519 | 7.42 
36 | 381528 | 5.68| .501131 | 7.50 || 36 | .401748 | 5.55 | ‘527964 | 7.42 
37 | 1381869] 5.70 | .501581 | 7.48 |} 37 | .402081 | 5.53 | .528409 | 7.40 
38 | 1382211 | 5.68} .502030 | 7.50 || 388 | .402413 | 5.53 | 1528853 | 7.42 
39 | 882552 | 5.68 | .502480 | 7.48 || 80] 402745 | 5.53 | 1529298 | 7.40 
40 | 1382893] 5.68 | 1502929 | 7.50 |) 40} .403077 | 5.53] 1529742 | 7.42 
41 | 9.383234 | 5.67 | 9.503379 | 7.48 || 41 | 9.403409 | 5.53 | 9.530187 | 7.40 
42 | 1388574. | 5.68] .503828| 7.48 || 42) 403741 | 5.53 | .530631 | 7.40 
43 | 1383915 | 5.67 | .504277 | 7.48 || 43] .404073 | 5.52] -531075 | 7.40 
44 | 1384255 | 5.67 | .504726| 7.48 || 44} .404404 | 5.53 | 1531519 | 7.40 
45 | 1384505 | 5.67 | .505175 | 7.48 || 45 | .404736 | 5:52 | 1531963] 7.40 
46 | 1384935] 5.67 | 1505624] 7.48 || 46 | 405067 | 5.52 | .582407 | 7.40 
7 | 1395975 | 5.67 | .506073 | 7.48 || 47] .405808 | 5.52] 582851 | 7.40 
48 | (385615 | 5.67 | .506522 | 7.48 || 48 | .405720 | 5.50 | -533295 | 7.40 
49 | 385955 | 5.65 | .506971 | 7.47 || 49 | .406059 | 5.58 | °538730 | 7.38 
50 | 386294} 5.67] 1507419 | 7.48 || 50 | .406390 | 5.52 | 534182 | 7.40 
51 | 9.986634 | 5.65 | 9.507868 | 7.47 || 51 | 9.406721 | 5.50 | 9.534686 | 7.40 
52 | 1386073 | 5.65 | .508316 | 7.48 || 52 | .407051 | 5.50 | 15385070 | 7.38 
53 | 1387312 | 5.65 | .508765 | 7.47 || 53 | 1407381 | 5.50 | 1585513 | 7.38 
54| 1887651 | 5.63 | 3500213 | 7.47 || 54 | .407711 | 5.50 | .535956 | 7.40 
55 | 1887989 | 5.65 | .509661 | 7.47 || 55 | .408041 | 5.50 | °536400 | 7.38 
56 | 1388328 | 5.63 | 1510109 | 7.47 || 56 | .408371 | 5.48 | .636843 | 7.38 
57 | 1388666 | 5.65 | .510557 | 7.47 || 57 | 408700 | 5.50 | .537286 | 7.38 
58} 1889005 | 5.63 | 1511005 | 7.47 || 58 | .409030 | 5.48] -537729 | 7.38 
59 | 1389843 | 5.63] 1511453 | 7.47 || 59 | 1409359 | 5.48 | 1538172 | 7.38 
60 | 9.889681 | 5.62] 9.511901 | 7.45 || 60 | 9.409688 | 5.48 | 9.538615 | 7.38 


en Un UIE Sn 


156 TABLE XXVI.—LOGARITHMIC VERSED SINES 




















42° 43° 
| 
4 Vers. PUDo rs) ixesecs yD. dan! Vers, | D.1".) Ex. 866.4 Do i:: 
0 | 9.409688 | 5.48 | 9.588615 | 7.38 O | 9.429181 | 5.35 | 9.565053 |. 7.32 
1 .410017 | 5.48 .939058 | 7.387 1 .429502 | 5.33 .565492 | 7.30 
2 .410346 | 5.48 .589500 | 7.38 2 .429822 | 5.33 .565930 | '7.32 
3 .410675 | 5.48 .539943 | 7.38 3 .430142 | 5.35 .566369 | 7.30 
4 .411004 | 5.47 .040386 | 7.37 4 .430463 | 5.383 .566807 | 7.30 
5 .4113832 | 5.47 .540828 | 7.38 5 .430783 | 5.33 .567245 | 7.30 
6 .411660 | 5.48 -541271 7.387 6 .431103 | 5.82 .567683 |. 7.30 
% | .411989 | 5.47 .041713 | 7.37 df -431422 | 5.33 .568121 80 
8 -412317 | 5.45 042155 | 7.37 8 .431742 | 5.33 .568559 30 
9 .412644 | 5.47 .542597 | 7.38 9 . 482062 | 5.32 .568997 30 
10 | 412972 | 5.47 | .543040 | 7.37 || 10 | .482381 | 5.82 | .569435 30 
11 | 9.413300 | 5.45 | 9.543482 37 || 11 | 9.432700 | 5.33 | 9.569873 30 
12 .413627 | 5.47 543924 Yeu tbe .483020 | 5.32 .570311 28 
13 .4138955 ; 5.45 544866 35 || 138 .433339 | 5.30 .570748 30 
14 .414282 | 5.45 .544807 37 || 14 .433657 | 5.32 .571186 30 


15 | .414609 | 5.45 | .545249 
16 -414986 | 5.45 | .545691 
17 | .415263 | 5.43 | .546132 
18 | .415589 | 5.45 | .54657 

19 | .415916 | 5.43 | .547015 
20 | .416242 |} 5.48] .547457 


21 | 9.416568 | 5.43 | 9.547898 
22] .416894 | 5.43 | .548339 
23 | .417220 | 5.43 | .548781 
24 | .417546 | 5.42 | 549222 
25 | .417871 | 5.43] .549663 
26 | .418197 | 5.42] .550104 
27 | .418522 | 5.43] .550544 


37 || 15 | .483976 | 5 32 | 571624 
35 || 16 | .484295 | 5.30] .572061 
37 || 17 | .484613 | 5.382 | .572498 
35 || 18} .484982 | 5.380 | .572936 
37 || 19} .485250 | 5.30] .573373 
2 .4385568 | 5.380 | .573810 


35 || 21 | 9.435886 | 5.30 | 9.574247 
87 || 22 | .486204 | 5.28 | .574685 
-30 || 23 .436521 | 5.30 | .575122 
.35 || 24 .436839 | 5.28 | .575558 
80 || 25 .487156 | 5.28 | .575995 
.33 || 26 .487473 | 5.30 | .576432 
.385 || 27) .487791 | 5.27 | .576869 








Sa aS HG ICS HS TES JS IS RS I HES BS IS Jee He thes She IES HS 
& 
@ 
bo Sis He Soc ES Ha Has Hogs Tas Hom Walaa ft Joos Slane Fees Is Ha 1S Es IS JRC IES FES 
ci) 
go 


28 | .418848 | 5.42 | .550985 .85 |) 28) .488107 | 5.28 | .577306 274 
29 | .419173 | 5.42 | .551426 .85 || 29 | .488424 | 5.28] 577742 28 
30 | .419498 | 5.40] .551867 | 7.33 |] 80 | .488741 | 5.28] .578179 27 
31 | 9.419822 | 5.42 | 9.552307 | 7.85 || 31 | 9.439058 | 5.27 | 9.578615 | 7.28 
32 | .420147 | 5.40 | .552748 | 7.33 || 832 | .4389374 | 5.27 | .579052 | 7.27 
33 | .420471 | 5.42] .553188 | 7.35 || 83] .4389690 | 5.28 | .579488 | 7.27 
34 | .420796 | 5.40 | .553629 | 7.33 || 34] .440007 | 5.27 | 579994 | 7.28 
35 | .421120} 5.40] .554069 | 7.33 || 35 | .440323 | 5.27 | .580361 | 7.27 
36 | .421444 | 5.40 | .554509 | 7.33 || 86 | .440639 | 5.25 | .580797 | 7.27 
37 | .421768 | 5.40] .554949 | 7.33 || 87} .440954 | 5.27 | .5812383 | 7.27 
38 | 422092 | 5.40 | .555389 | 7.33 || 88] .441270 | 5.25 | .581669| 7.27 
39 | .422416 | 5.38 |~ .555829 | 7.33 || 89 | .441585 | 5.27] .582105| 7.27 
40 | .4227389 | 5.40] .556269 | 7.33 || 40| .441901 | 5.25 | .582541 | 7.27 
41 | 9.423063 } 5.38 | 9.556709 | 7.33 || 41 | 9.442216 | 5.25 | 9.582977 | 7.2 
42 | 423386 | 5.388] .557149 | 7.83 || 42 | .442531 | 5.25 | .5834138 | 7.25 
43 | .423709 | 5.38] .557589 | 7.82 || 43 | .442846 | 5.25] .583848 | 7.97 
44 | 424082 | 5.388 | .558028 | 7.33 || 44] .448161 | 5.25 | .584284 | 7.97 
45 | 424855 | 5.387] .558468 | 7.82 || 45 | .443476 | 5.23 | .584720 | 7.25 
46 | .424677 | 5.38] .558907 .33 || 46 | .443790 | 5.25 | .585155 | 7,27 
47 | .425000 | 5.37 


48 | .425322 | 5.38] .559786 .33 || 48 | .444419 | 5.2 586026 | 7.2 


7 
559347 (.88 47 | .444105 | 5.23 | .585591 | 7.25 
‘ 
49 | .425645 | 5.37] .560226 | 7.32 || 49 | 444738 | 5.23 | .586462.| 7.25 





Ot OC OT CO STOUT Oe 





50 | .425967 | 5.37 | .560665 | 7.32 || 50 | .445047 | 5.23 | 586807 | 7.25 
51 | 9.426289 | 5.37 | 9.561104 | 7.82 || 51 | 9.445361 | 5.23 | 9.587382 | 7.2 
52 | 426611 | 5.87 | 1561543 | 7.82 || 52 | 445675 | 5.23 | 587767 | 712 
53 | .426933 | 5.35 | 561982 | 7.32 || 53| .445989 | 5.22 | ‘588203 | 7.2 
54] 427254 | 5.87] .562421 | 7.32 || 54 | 446302 | 5.93 | 685638 | 7.2 
55 | .427576 | 5.35 | .562860 | 7.32 || 55 | .416616 | 5.22 1589073 | 7.2 
56 | .427897 | 5.85 | .568209] 7.32 || 56 | 1446929 | 5.22} 589508 | 7.2 
BY | .428218 | 5.85 | .563738| 7.30 || 57 | .447242 | 5.22] 589042 | 7.2 
58 | .428539 | 5.85 | 1564176 | 7.32 || 58] 447555 | 5.92] 590377 | 72 
59 |, .428860 | 5.35 | 1564015 | 7.30 || 59 | (447868 | 5.22 | 590812 | 7.2 
60 | 9.420181 | 5.33 | 9.565053 | 7.32 || 60 | 9.448181 | 5.20 | 9.591247 | 7.23 


a 


— Se 


AND EXTERNAL SECANTS 157 



































44° 45° 

4 Vers: MieDialt ix sec. D: 1": , Vers. |D. 1".| Ex. sec. |D. 1’. 
0 | 9.448181 5.20 | 9.591247 23 0 | 9.466709 | 5.08 | 9.617224 7.20 
al .448493 5.22 .591681 GE 25 1 .467014 | 5.08 .617656 7.18 
2 . 448806 5.20 .592116 25 2 .467319 | 5.08 . 618087 7.18 
3 .449118 5.22 .592551 7.23 3 .467624 | 5.07 .618518 7.18 
4 449431 5.20 .592985 % 23 4 .467928 | 5.08 .618949 7.18 
5 449743 5.20 .593419 Vaz 5 468233 | 5.07 .619380 7.18 
6 .450055 5.18 .598854 1323 6 468537 | 5.07 .619811 7.18 
” .450366 5.20 .594288 7.23 f¢ 468841 | 5.07 . 620242 7.18 
8 .450678 5.20 .594722 7.23 8 469145 | 5.07 .620673 7.18 
9 450990 5.18 .595156 25 9 .469449 | 5.07 .621104 eles: 
10 451301 | 5.18 .595591 | 7.23 || 10 .4697538 | 5.07 .621535 | 7.18 
11 | 9.451612 5.20 | 9.596025 7.23 || 11 | 9.470057 | 5:05.) 9.621966 fee 
ehe .451924 5.18 .596459 (25 12 .470360 | 5.07 . 622396 7.18 
13 ~ 452235 5.18 .596893 7 22 13 .470664 | 5.05 . 622827 7.18 
14 .452546 Daly .597326 7.23 14 .470967 | 5.05 .628258 7 1G 
15 452856 Dele, .597760 7.23 15 .471270 | 5.05 . 628688 7.18 
16 .453167 5.18 .598194 (e258 16 401573! | 5-05 .624119 TAG 
i876 .453478 517 .598628 Wa ve .471876 | 5.05 . 624549 7.18 
18 .453788 5.17 .599061 econ dS 2472179 | 5.05 .624980 ea 
19 .454098 57 .599495 (22 11 19 .472482 | 5.03 . 625410 7.18 
20 .454408 IS} aLf¢ .599928 7.23 || 20 6472784 | 5.05 .625841 ela 
21 | 9.454718 5.17 | 9.600362 7.22 || 21 | 9.473087 | 5.03 | 9.626271 (bear 
22 .455028 | 5.17 .600795 7223) 99 .473889 | 5.03 .626701 cama lf 
23 6455338 5.17 .601229 22 Wi 23 .473691 | 5.038 .627131 CNG 
24 .455648 5.15.9" [601662 eae e4 .4738993 | 5.03 .627561 CRG 
53) .455957 Dale .602095 % 22 || 25 474295 | 5.03 .627991 iat if 
26 .456267 DatD . 602528 Micon eG .474597 | 5.08 . 628421 iG 
27 .456576 5.15 . 602962 PBA Ose .474899 | 5.02 .628851 (cali 
28 .456885 5.15 . 60383895 7.22 ||' 28 .475200 | 5.038 .629281 hoe 
29 .457194 5p .603828 7.22 || 29 .475502 | 5.02 .629711 ea bfs 
30 -457503 5.13 |: .604261 7.22 || 30 .475808 | 5.02 .630141 TOG 
31 | 9.457811 5.15 | 9.604694 7.20 || 31 | 9.476104 | 5.02 | 9.680571 Gee fs 
32 .458120 5.15 .605126 (nee. [| 32 .476405 | 5.02 .631001 “elo 
33 458429 5.13 . 605559 (Re2alPoo .476706 | 5.02 .631430 en Wy 
34 458737 5.13 . 605992 (haere |e 477007 | 5.02 .631860 eae 
35 .459045 5.13 . 606425 7.20 || 35 .477308 | 5.00 .632290 aes: 
36 .459853 &.13 . 606857 7.22 || 36 .477608 | 5.02 . 6382719 (oak 
37 .459661 5.13 .607290 4.20 |) 37 .477909 | 5.00 .633149 Wels 
38 .459969 5.13 607722 (22 | 38 .478209 | 5.00 . 638578 OG 
39 .46027 Bake . 608155 7.20 || 39 .478509 | 5.00 . 634008 7.15 
40 .460584 5.13 .608587 7.22 || 40 .478809 | 5.00 .634437 WeLo 
41 | 9.460892 | 5.12 | 9.609020 | 7.20 || 41 | 9.479109 | 5.00 | 9.634866 | 7.17 
42 .461199 D.12 . 609452 7.20 || 42 .479409 | 5.00 . 635296 (ols 
43 .461506 D.12 . 609884 7.20 || 43 .479709 | 5.00 .635725 Tels 
44 .461813 5.12 .610316 7.22 || 44 .480009 | 4.98 . 636154 7.15 
45 .462120 5.12 .610749 7.20 || 45 .480308 | 5.00 .636583 lags 
46 -46242 5.12 .611181 7.20 || 46 .480608 | 4.98 .637012 Vale 
47 4627, 5.10 .611613 7.20 re .480907 | 4.98 .637441 V.15 
48 .463040 5.12 .612045 7.20 || 48 .481206 | 4.98 .637870 7.15 
49 .463347 5.10 .612477 7.18 |; 49 .481505 | 4.98 .688299 7.15 
50 .463653 5.10 .612908 7.20 || 50 .481804 | 4.98 .638728 els: 
51 | 9.463959 | 5.10 | 9.613340 |} 7.20 || 51 | 9.482103 | 4.97 | 9.639157 } 7.15 
52 .464265 5.10 .613772 Peer) IN a4 .482401 | 4.98 . 639586 ols 
53 .464571 5.10 ly. 614204 7.18 || 53 .482700 | 4.97 .640015 7.13 
54 .464877 5.10 .614635 7.20 || 54 .482998 | 4.97 . 640443 Wel 
55 .465183 5.08 .615067 7.20 || 55 .483296 | 4.98 .640872 710 
56 .465488 5.10 . 615499 7.18 || 56 .483595 | 4.97 .641301 138 
57 .465794 5.08 . 615930 7.20 || 57 .483893 | 4.97 .641729 og) 
58 .466099 5.08 .616362 7.18 || 58 .484191 | 4.95 .642158 7.13 
59 .466404 5.08 .616793 7.18 || 59 .484488 | 4.97 . 642586 {ipa bs) 
60 | 9.466709 5.08 | 9.617224 7.20 |} 60 | 9.484786 | 4.97 | 9.643015 7.13 








15 


TABLE XXVI.—LOGARITHMIC VERSED SINES 

















Vers. 

0 | 9.484786 
1 .485084 
2 .485381 
3 .485678 
4 .485976 
5 .486273 
6 .486570 
q .486866 
8 .487163 
9 .487460 
10 .487756 
11 | 9.488053 
12 .488349 
13 .488645 
14 .488941 
15 .489237 
16 .489533 
17 .489828 
18 .490124 
19 -490119 
20 .490714 
21 | 9.491010 
22 .491305 
23 .491600 
24 .491894 
25 .492189 
26 .492484 
2 492778 
28 .498072 
29 .493367 
30 .493661 
31 | 9.493955 
32 .494249 
33 .494542 
34 .494836 
35 .495130 
36 .495423 
"e .495716 
38 .496009 
39 . 496802 
40 .496595 
41 | 9.496888 
42 -497181 
43 497473 
44 .497766 
45 .498058 
46 -498350 
47 .498643 
48 -498935 
49 -499226 
50 .499518 
51 | 9.499810 
52 .600101 
53 . 500398 
54 . 500684 
55 .500975 
56 .501266 
57 . 501557 
58 .501848 
59 502139 


60 | 9.502429 





46° 


D. 1", |Ex. sec. 





wrowMwwoyo 
CO Ot Ove OU 


SCWOCWWOWWD Bint ewwwwwe St oo ot ot oe 


S 


DOHDDDOSCDS 


@ 
GO 


4.88 


AAA ALDARA Ae 
(o oie offe offe oe sho oe ofe ofc oe ome oe fe ofe 2) 
f Orcs Or O1rorer sr Ot ROT NINN 








647297 
9.647725 
648153 
648581 
649009 
649436 
649864 
650292 
650720 
651147 
651575 
9.652002 
652430 
“652857 
653285 
653712 
654140 
654567 
654994 
655421 
655849 


9.656276 
656703 
657130 
657557 
657984 
658411 
.658833 
659265 
-659691 
.660118 


9.660545 
. 660972 
.661398 
-661825 
662252 
.662678 
. 663105 
663531 
663958 
-664384 


9.664810 
665237 
665663 
666089 
.666515 
666942 
.667368 
667794 
668220 


9.668646 





~ 


AE EAE AS AE AE EA FI -2 


fas et Se ee ee es ew a he ME EES Sr, 


a3 AB AI JAPA 


aI AF aT AIA YI-I-I- 


MINI 





— Re 
ww GW Co 


— 
qo co to 



































6 ot 


aX 


47° 
/ Vers. D. 1". | Ex. see. |D. 1°. 
0 | 9.502429 | 4.85 | 9.668646 7.10 
1 .502720 | 4.838 , 669072 7.10 
2 .503010 | 4.838 . 669498 7.10 
3 .503800 ; 4.85 . 669924 7.10 
4 .503591 | 4.838 . 670850 YO) 
By .503881 | 4.83 670776 7.08 
6 .504171 | 4.82 .671201 7.10 
7 .504460 | 4.83 .671627 7.10 
8 .504750 | 4.83 . 672053 7.10 
9 .505040 | 4.82 .672479 7.08 
10 .505329 | 4.82 .672904 | 7.10 
11 | 9.505618 | 4.83 | 9.673330 | ‘7.10 
12 .505908 | 4.82 673756 7.08 
13 506197 | 4.82 .674181 7.10 
14 .506486 | 4.82 .674607 7.08 
15 .506775 | 4.80 . 6750382 yen tb) 
16 .507063 | 4.82 , 675458 7.08 
1% .507352 | 4.80 ,675883 | 7. 
18 .507640 | 4.82 . 6763809 7.08 
19 . 507929 | 4.80 676784 7.08 
20 .508217 | 4.80 .677159 7.08 
21 | 9.508505 | 4.80 | 9.677584 | 7.10 
22 .508793 | 4.80 .678010 "08 
23 .509081 | 4.80 , 6784385 7.08 
24 .509369 | 4.80 .678860 7.08 — 
25 -509657 | 4.80 , 679285 7.08 
26 -509945 | 4.7 679710 7.10 
27 510282 | 4.80 .680136 | 7.08 
28 .510520 | 4.78 . 680561 7.08 
29 .510807 | 4.78 .680986 | 7. 
30 .511094 | 4.78 .681411 | 7.08 
81 | 9.511381 | 4.78 | 9.681886 | 7.07 
82 .511668 | 4.78 .682260 | 7.08 
555) -511955 | 4.77 .682685 7.08 
34 .512241 | 4.78 . 688110 7.08 
85 .612528 | 4.78 . 683535 7.08 
36 -512815 | 4.77 .683960 7.08 
37 -613101 |). 4.77 .684385 7.07 
38 .618887 | 4.77 . 684809 7.08 
89 .613673 | 4.7 - 685234 7.08 — 
40 .513959 | 4.77 .685659 | 7.07 
41 | 9.514245 | 4.77 | 9.686088 |) 7.08 
42 -614581 | 4.77 .686508 7.08 
43 -514817 | 4.75 . 686933 7 OCm 
44 ALSTOR. | 4er7 - 687357 7.08 | 
45 .5153888 | 4 75 . 687782 T. 
46 -615673 | 4.77 . 688206 yes 
ik .515959 | 4.75 .688631 eis 
48 516244 | 4:7 . 689055 7 
49 .516529 | 4.75 .689479 |. 
50 .516814 | 4.73 . 689904 ‘es 
51 | 9.517098 | 4.75 | 9.690828 | 7. 
52 .5173883 | 4.75 .690752 it 
53 .517668 | 4.73 -691177 ie 
54 .517952 | 4.738 .691601 file 
55 .518236 | 4.75 . 692025 he 
56 -518521 | 4.73 .692449 % 
57 .518805 | 4.73 - 692873 ke 
58 .519089 | 4.7 . 693298 a: 
: 519373 | 4.78 698722 : ; 





9.519657 


9.694146 





|}oocsoooce ¢ 
S23S2293383383S SSSssss 


AND EXTERNAL SECANTS 1159 




































































48° :; 49° 
| 
’ Vers. Dei tixasec: | DS1", a Vers.- | D. I'./7 Ex. see. |D. 1". 
0 | 9.519657 | 4.72 | 9.694146 | 7.07 0 | 9.586484 | 4.62 | 9.719541 | 7.05 
1 .519940 | 4.73 FO945 70) | a) FOG meet .586761 | 4.62 719964 | 7.05 
2 .520224 |} 4.721 .694994 1 7.07 2 .587088 | 4.62 720886 | 7.05 
3 -520507 | 4.73 .695418 | 7.07 8 .537315 | 4.62 - 720809 | 7.03 
4 520791 4 4.72 .695842 | 7.07 4 .537592 | 4.62 2123003 
5 sB2i4 |} Aye .696266 | 7.05 5 .5387869 | 4.60 -721653 | 7.05 
6 -521857 | 4.72 .696689 | 7.07 6 -588145 | 4.62 722076 | 7.03 
ff .521640 | 4.7 697113 | 7.07 i .588422 | 4.60 722498 | 7.05 
8 25210095) 1457: .6975387 | 7.07 8 .538698 | 4.60 722921 7.03 
9 .522206 | 4.70 .697961 | 7.07 9 .538974 | 4.62 3723348 | 17103 
10 .522488 | 4.72 .698885 | 7.07 |; 10 .589251 | 4.60 728765 | 7.05 
11 | 9.522771 | 4.72 | 9.698809 | 7.05 || 11 | 9.539527 | 4.60 | 9.724188 | 7.03 
12 .523054 | 4.70 .699282 | 7.07 || 12 .539803 | 4.60 724610 | ‘7.03 
13 523836 4.70 .699656 7.07 13 .540079 | 4.58 (25082 7.08 
14 .523618 | 4.70 -700080 | 7.05 || 14 .540354 | 4.60 725454 | 7.05 
15 .528900 | 4.70 . 700503} 7.07 || 15 .540630 | 4.60 2425877. | 7203: 
16 .624182 | 4.70 .700927 | 7.05 || 16 .540906 | 4.58 .726299 | 7.03 
ag 524464 | 4.70 S4O1850: | C07 1077 .541181 | 4.58 N26721_ |-27203 
18 -524746 | 4.70 701774 | 7.07 || 18 .541456 | 4.60 M2143. | 17603 
19 .525028 4.68 . 702198 7.05 19 .541732 | 4.58 727565 7.05 
20 .525309 | 4.70 . 702621 7.07 || 20 .542007 | 4.58 727988 | 7.03 
21 | 9.525591 | 4.68 | 9.703045 | 7.05 || 21 | 9.542282 | 4.58 | 9.728410 | 7.03 
22 52587 4.68 -708468 | 7.05 || 22 .542557 | 4.58 .728882 | 7.03 
23 -626153.| 4. % 703891 7,07 || 28 .542832 | 4.57 729254.) 72038 
24 -526485 | 4.68-| .704815 | 7.05 || 24 .548106 | 4.58 .729676 | 7.08 
25 .526716 | 4.68 704738 | 7.07 || 25 .548881 | 4.57 .730098 | 7.03 
26 .526997 | 4.67 . 705162 | 7.05 || 26 .548655 | 4.58 N7B05200 | es 
27 527277 | 4.68 ~(O5585 |) 7805 |) 27 .548930 | 4.57 -130042 | 7:08 
28 .527558 |. 4.68 -706008 | 7.05 || 28 .544204 | 4.57 nroLe0d, || 103 
29 -527839 | 4.67 -706431 7.07 1029 .544478 | 4.57 OT7S6. | E038 
30 .528119 | 4.68 -706855 | 7.05 || 30 544752 | 4.57 - 7382208 | 7.03 
31 | 9.528400 | 4.67 | 9.707278 | 7.05 |) 31 | 9.545026 | 4.57 | 9.732630 | 7.03 
2 -528680 | 4.67 707701 | 705 || 382 .545800 | 4.57 3180052) | 72038: 
33 -528960 | 4.67 . 708124 |- 7.05 || 38 .545574 | 4.57 133474) V7E03: 
34 -529240 | 4.67 (08547 | 7.07 || 34 .545848 | 4.55 . 738896 | 7.02 
35 .529520 | 4.67 - 708971 7.05 || 35 .546121 | 4.57 34817 | 7.03 
36 .529800 | 4.67 .709394 | 7.05 || 36 .5463895 | 4.55 . 734739 | 7.03 
37 .530080 |. 4.65 709817 | 7.05 || 37 .546668 | 4.55 . 7385161 7.03 
38 .5803859 | 4.67 710240 | 7.05 || 88 .546941 | 4.55 7855838 | 7,038 
39 .580639 | 4.65 . 710668 |. 7.05 || 389 .547214 | 4.55 .736005 | 7.03 
40 .580918 | 4.67 -711086 | 7.05 || 40 .547487 | 4.55 7386427 | 7.02 
41 | 9.531198 | 4.65 | 9.711509 | 7.05 || 41 | 9.547760 | 4.55 | 9.736848 | 7.03 
42 .681477 | 4.65 .711982:| 7.05 || 42 .548038 | 4.55 737270 | 7.03 
43 .531756 | 4.65 712855 | 7.05 || 43 .548806 | 4.55 .7387692 | 7.03 
44} .532085 | 4.65 .712778 | 7.03 || 44 .548579 | 4.53 .738114 | 7.02 
45 .582314 | 4.63 .718200 | 7.05 || 45 .548851 | 4.55 .738535 | 7.08 
46 .532592 | 4.65 . 718623 | 7.05 || 46 549124 | 4.53 738957 | 7.08 
ii .5382871 | 4.65 -(14046 | 7.05 || 47 .549396 | 4.53 .739879 | 7.02 
48 .583150 | 4.63 . 714469 | 7.05 || 48 .549668 | 4.53 .739800 | 7.03 
49°} .533428 | 4.63 14892 | 7.05 || 49 .549940 | 4.53 740222 | 7.03 
50 -5388706 | 4.65 . 715815 |. 7.03 || 50 .550212 | 4.53 740644 | 7.02 
51 | 9.533985 | 4.63 | 9.715787 | 7.05 || 51 | 9.550484 | 4.53 | 9.741063 | 7.03 
52 .584263 | 4.63 -716160 | 7.05 || 52 .550756 | 4.53 741487 | 7.02 
53 6384541 4.63.; .716588 | 7.08 || 53 .551028 | 4.52 .%41908 | 7.08 
54 .5384819 | 4.63 - 717005 | 7.05 || 54 .551299 | 4.53 742330. | 7.02 
55 .585097 | 4.62 717428 | 7.05 ‘|| 55 {551571 -| 4°52 742751 | 7.08 
56 .5353874 | 4.63 .717851 | 7.03 || 56 .551842 | 4.52 748178. | 72038 
57 .5385652 | 4.62 . 718273 | 7.05 || 57 .652113 | 4:52 | .748595 | 7.02 
58 .585929 | 4.63 .718696 | 7.03 || 58 .552384 | 4.53 .744016 | 7.08 
59 . 536207 4.62 . 719118 7.05 59 .552656 | 4.52 744438 7.02 
60 | 9.536484 |. 4.62 | 9.719541 7.05 || 60 | 9.552927 | 4.50 | 9.744859 | 7.02 





160 TABLE XXVI.—LOGARITHMIC VERSED SINES 


51° 





| 
WA Vers) eed hy Eexeseei DEL. , WVerss 1D.’ viikext secu Des 





| | | 





9.568999 | 4.42 | 9.770127 

















0 | 9.552927 | 4.50 | 9.744859 | 7.02 0 7.02 
1 .553197 | 4.52 | .745280 | 7.03 1 .569264 | 4.40 | .770548 | 7.02 
(2 | 553468 | 4.52 ~745702 | 7.02 2| .569528 | 4.42 | .770969 | 7.00 
3} .5538739 | 4.50] .746123 | 7.03 3} .569793 | 4.40 | .771389 } 7.02 
4) .554009 | 4.52] .746545 ) 7.02 4| .570057 | 4.42 | ..771810} 7.02 
5 | .554280 |] 4.50! .746966 | 7.03 5 | .570822 | 4.40 | .772231 | 7.02 
6 | .554550 | 4.50] .7473888 | 7.02 6 | .570586 | 4.40 | .772652 | 7.02 
7 | .554820 | 4.52) .747809 | 7.02 7 | 570850 | 4.40 | .773073 | 7.02 
8 | .555091 |} 4.50 | .748230 | 7.03 8 | .571114 | 4.40 | .7738494 | 7.00 
9 | .555361 | 4.50) .748652 | 7.02 9 .571878 | 4.40-| .773914 | 7.02 
10 | .555681 | 4.48 | .749073 | 7.02 || 10 | .571642 | 4.40 | .774885 | 7.02 
11 | 9.555900 | 4.50 | 9.749494 | 7.03 |} 11 | 9.571906 | 4.40 | 9.774756 | 7.02 
12 | .556170 | 4.50] .749916 | 7.02 || 12 572170 | 4.40 | .7%5177 | 7.02 
13 | .556440 | 4.48 | .750337 | 7.02 || 13 572434 | 4.88 | .775598 | 7.00 
14} .556709 | 4.50 (50758 | 7.03 || 14] .572697 | 4.88 | .776018 | 7.02 
15 | .556979 | 4.48} .751180 | 7.02 || 15 .572960 | 4.40 | .776489 | 7.02 
16} .557248 | 4.48 |} .751601 | 7.02 || 16 573224 | 4.88 | .776860 | 7.02 
17 | .557517 | 4.48 752022 | 7.02 || 17 | .573487 | 4.38 | .777281 | 7.02 
18 |. .557786 | 4.48 | .752443 | 7.03 || 18] .573750 | 4.388 | .777702 | 7.00 
19 | .558055 | 4.48 |} .752865 | 7.02 || 19 .574013 | 4.88 | .778122 | 7.02 
20 | .5583824 | 4.48 | .7538286 | 7.02 || 20 .574276 | 4.38 | .778543 | 7.02 
21 | 9.558593 | 4.48 | 9.753707 | 7.02 || 21 | 9.574539 | 4.38 | 9.778964 | ~7.02 
22 | .558862 | 4.48 | .754128 | 7.02 || 22 .574802 | 4.37 779385 | 7.00 
23 559131 | 4.47 754549 | 7.03 || 23} .575064 | 4.388 | .779805 | 7.02 
24 559399 | 4.47 754971 | 7.02 || 24 5753827 | 4.87 | 780226 | 7.02 
25 559667 | 4.48 755392 | 7.02 || 25 .575589 | 4.88 | .780647 | 7.02 
26 559936 | 4.47 755813 | 7.02 || 26] .575852 | 4.37 | .781068 | 7. 
27 560204 | 4.47 75623 7.02 || 27 576114 | 4.87 | .781488 | 7. 
28 56047 4.47 756655 | 7.02 || 28 -576376 | 4.87 | .781909 | 7. 
29) .560740 | 4.47 | .75707 7.03 || 29 .576688 | 4.37 | .782830 | 7. 
80 |} .561008 | 4.47 | .757498 | 7.02 || 80] .576900 | 4.87 | .782751 | 7. 
31 | 9.561276 | 4.47 | 9.757919 | 7.02 || 31 | 9.577162 | 4.87 | 9.783171 | 7. 
82 | .561544 | 4.45 | .758340 | 7.02 || 32 577424 | 4.385 |] .788592 | 7. 
83 | .561811 | 4.47 | .758761 | 7.02 || 83] 577685 | 4.387 | .7840138 | 7. 
84 | .562079 | 4.45 759182 | 7.02 || 84] .577947 | 4.385 | .784483 | 7. 
35 | .562846 | 4.45 | .759603 | 7.02 {| 85 | .578208 | 4.37 | .784854] 7. 
36 | .5626138 | 4.47) .760024 | 7.02 || 86 | .578470 | 4.35 | 785275 | 7. 
37 | .562881 | 4.45 | .760445 | 7.02 || 37 | .5787381 | 4.35 |. .785696 | 7. 
88 | .563148 | 4.45 760866 | 7.02 || 88] .578992 | 4.385 | .786116 | 7. 
39 | .563415 | 4.45 | 1.761287 | 7.02 || 389 | .579253 | 4.85 | .7865387 | 7. 
40 | .563682 | 4.43 | .761708 | 7.02 || 40 | .579514 | 4.35 |] .786958] 7. 
41 | 9.563948 | 4.45 | 9.762129 | 7.02 || 41 | 9.579775 | 4.35 | 9.787378 | 7.0 
42 | .564215 | 4.45] .762550 | 7.02 || 42) .580086 | 4.385) .787799 | 7.0 
43 | .564482! 4.43 | .762971 | 7.02 || 43 | .580297 | 4.33 788220 | 7.0 
44 | .564748 | 4.45 | .768892 | 7.02 || 44 | .580557 | 4.35 788641 | 7.0 
45 | .565015 | 4.48 |) .7638138 | 7.02 || 45 | .580818 | 4.33 | .789061 | 7.0 
46 | .565281 | 4.43 (64234 | 7.02 || 46 581078 | 4.35 789482 | 7.0 
47 | .565547 |- 4.438 | .764655 | 7.02 || 47 | .581839 | 4.33 789903 | %. 
48) .565813 | 4.43 | .765076 | 7.02 || 48} .581599 | 4.33 | .790823 | 7.0: 
49 | .566079 | 4.43 | .765497 | 7.02 || 49 | .581850 | 4.33 | .790744 | 7.0 
50 | .5663845 | 4.43 | .765918 | 7.02 || 50 | .582119 | 4.33 | .791165 | 7.0 
51 | 9.566611 | 4.43 | 9.766339 | 7.02 || 51 | 9.582379 | 4.33 | 9.791586 | 7.0 
52 | .566877 | 4.42 766760 | 7.02 || 52] .582639 | 4.382 | .792006 | 7.0 
53 | .567142 | 4.48 | .767181 | 7.02 || 53 | .582898 |} 4.338} .792427 | 7.0 
54 | .567408 | 4.42 | .767602 | 7.00 || 54] .583158 | 4.33] .792848 | 7.0 
5D | .567673 | 4.42] .768022 | 7.02 || 55 |} .583418 | 4.32) .793268 | 7.0: 
56 | .567938 | 4.43 | .768443 | 7.02 || 56] .583677 | 4.32 | .793689 | 7.0: 
57 | .568204 | 4.42) .768864 | 7.02 || 57 | 5839386 | 4.383 | .794110} 7.0 
58 | .568469 | 4.42) .769285} 7.02 || 58 | .584196 | 4.32] .794531 ] 7.0 
59 | ..5687384 | 4.42] .769706 | 7.02 || 59) .584455 | 4.32 |] .794951 | 7. 
60 | 9.568999 | 4.42 | 9.770127 | 7.02 || 60 | 9.584714 | 4.32 | 9.795372 | 7. 


f=) Oofcosaoafo os .ofo 60> 
WNWOCWNWNVOWOVWS SSSSSBSSESES SSSSESESRES SS=ESS 





AND EXTERNAL SECANTS 161. 
































9.602866 | 4.22 | 9.825254 
.608119 | 4.20 | .825675 
.603371 | 4.20 | .826096 
-603623 | 4.20] .826517 
.603875 | 4.20 | .826938 
.604127 | 4.20 827360 

2 || 17 .604879 | 4.20 | 827781 

02 || 18 .604631 | 4.20 | .828202 

02 || 19 -604883 | 4.18 | .828623 

00 || 20 .6051384 | 4.20 | .829044 


02 || 21 | 9.605386 | 4.18 | 9.829466 
z || 22 .605637 | 4.18 | .829887 
23 | .590646 | 4.28 805049 02 || 23 | - .605888 | 4.20 | .830308 
24] .590903 | 4.28 |. .805470 02 || 24 .606140 | 4.18 8380729 
25) .591160 | 4.27 .805891 | 7.00 || 25 .606391 | 4.18 | .881151 
26 |} .591416 | 4.28 .806311 | 7.02 || 26 .606642 | 4.18 | .831572 
27 | .591673 | 4.27 .806782 | 7.02 || 27 .606893 | 4.18 831993 
28 | -.591929 | 4.27 807153 .02 || 28 607144 | 4.17} .882415 
299 | .592185 | 4.28 807574 ’.02 || 29 607394 | 4.18 832836 


11 | 9.587557 | 4.30 | 9.800000 

2] .587815 | 4.380; 800421 
18 | .588073 | 4.80 800841 
14 | .588331 | 4.28 801262 
15 |. .588588 | 4.28] .801683 
16 | .588846 | 4.28 802104 
17 | .589103 | 4.80] .802524 
18 | .589361 | 4.2 802945 
19 | .589618 | 4.28 803366 
20 | .589875 | 4.28] .803787 


21 | 9.590182} 4.28 | 9.804207 
22 | .590889-| 4.2 804628 


Soo9o & 


52° 53° 
, Verseaiab. 13, kx secs | Del ssl)! Vers. |D.1’.| Ex. sec. | D. 1”. 
0 | 9.584714 4.32 | 9.795372 % 02 0 | 9.600085 | 4.22 | 9.820622 7.02 
1 .584973 4.32 7957938 7.00 1 .600838 | 4.22 . 8210438 7.02 
2 . 585232 4.32 . 796213 02 2 600591 | 4.238 . 821464 7.02 
3 .885491 4.30 . 796634 7.02 3 -600845 | 4.22 821885 (02) 
4 .585749 | ° 4.32 ~ 797055 7.02 4 .601098 | 4.22 - 822306 7.02 
5 .586008 4.30 97476 7.00 5 6013851 | 4.20 822727 7.02 
6 . 586266 4.32 . 797896 7.02 6 .6016038 | 4.22 .823148 7.02 
6 .586525 4.30 . 798317 7.02 { -601856 | 4.22 . 823569 7.02 
8 .586783 4.30 . 798738 7.00 8 .602109 | 4.22 . 823990 7.02 
9 .587041 4.30 .799158 7.02 9 . 602362 | 4.20 .824411 7.03 
10 .587299 4.30 .799579 02°}; 10 .602614 | 4.20 . 824833 like 
alal 
12 
13 
14 
15 
16 


Ks) 
WW WMWMWWW W 


fee Soe hes Ses Se es Se hs he 
S 
~] 
4 
“] 


AI-I-IW-X 
ri 





Pd ce ee ee ee a ee ee 
ci) 


7 

‘ 
80 | 592442 | 4.27 807995 | 7.00 || 80] .607645 | 4.18 | .838257 03 
81 | 9.592698 | 4.27 | 9.808415 | 7.02 || 31 | 9.607896 | 4.17 | 9.833679 | 7.02 
82 | .592954 | 4.27] .808836 | 7.02 || 82} .608146 |; 4.18 | .8384100 | 7.08 
83 | .5938210 | 4.27 .809257 | 7.02 || 83 | .608897 | 4.17 | .834522 | 7.02 
34 | .593466 | 4.25 .809678 | 7.02 || 34 .608647 | 4.17 | .884943 | 7.02 
35 | .598721 | 4.27 .810099 | 7.02 || 35 .608897 | 4.17 | .8385864 | 7.03 
386 | .5938977 |. 4.27 .810520 | 7.00 || 36 .609147 | 4.17 | .885786 | 7.02 
87 | .594238 | 4.25 .810940 | 7.02 || 37 | .609897 | 4.17 | .886207 | 7.03 
388 | .594488 | 4.25 811361 | 7.02 || 38 .609647 | 4.17 | .886629 | 7.02 
39 | .594748 | 4.27 -811782 | 7.02 || 39 .609897 | 4.17 | .887050 | 7.03 
40 |} .594999 | 4.25 6812203 | 7.02 || 40 | .610147 | 4.17 | 887472 | 7.02 

7 

7 

v4 

: 

















41 | 9.595254 | 4.25 | 9.812624 .02 || 41 | 9.610397 | 4.15 | 9.887893 | 7.03 
42 | .595509 | 4.25 813045 02 || 42 |] .610646 | 4.17 | .838315 | 7.02 
43 | .595764 | 4.25 813466 .00 || 43 | .610896 | 4.15 .838736 | 7.03 
44 | .596019 |} 4.25 818886 .02 || 44 .611145 | 4.15 | .839158 | 7.02 
45 | .596274 | 4.23 .814307 | 7.02 || 45 | .611894 | 4.17 | .839579 | 7.03 
46 | .596528 | 4.25 .814728 | 7.02 || 46 | .611644 | 4.15} .840001 | 7.08 
47 | .596783 | 4.25 .815149 | 7.02 |) 47 | .611893 | 4.15 | .840423 | 7.02 
48 | .597038 | 4.23 .815570 | 7.02 || 48 .612142 | 4.15 | .840844 | 7.063 
49 | .597292 | 4.23 .815991 | 7.02 || 49 |} .612391 | 4.15 | .841266 | 7.02 
50, .597546 |. 4.25, .816412 | 7.02 || 50) .612640 | 4.13 | 841687 , 7.03 
51 | 9.597801 | 4.23 | 9.816883 | -7.02 || 51 | 9.612888 | 4.15 | 9.842109 | 7.03 
52 598055 | 4.23 817254 | 7.02 || 52} .6181387 | 4.15 842531 | 7.03 
53 |  .598309 | 4.239) .817675 | 7.02 || 53 | .618386 | 4.138 | .842953 | 7.02 
54} .598563 | 4.23 .818096 | 7.02 || 54 .618684 | 4.15 .848374 | 7.03 
55 | .598817 | 4.23 .818517 | 7.02 |) 55 | .613883 | 4.138 | .848796 | 7.03 
56 | .599071 | 4.22 .818938 | 7.02 || 56 .614131 | 4.13 | .844218 | 7.02 
57 | .599824 | 4.23 .819359 | 7.02 || 57 | .614879 | 4.138 | .844639 | 7.03 
58 | .59957 4.22 .819780 | 7.02 || 58} .614627 | 4.15 | .845061 | 7.03 
59 | .599831 | 4.22 | .820201 |. 7.02 || 59 | .614876 | 4.13 | .845483 | 7.08 
60 | 9.600085 | 4.22 | 9.820622 | 7.02 |] 60 | 9.615124 | 4.12 | 9.845905 | 7.03 








i 


162 TABLE XXVI.—LOGARITHMIC VERSED SINES 
RMN ot 5 Fee ee Ee TN eee 


54° 55° 

















Q 








, Vers. | D. 1". | Ex. sec. | D. 1”. |} 7” Vers. |D.1".| Hx. sec. |D. 1". 
O | 9.615124 4.12 | 9.845905 7.03 0 | 9.629841 | 4.05 | 9.871250 7.05 
1 .615371 4.13 .846327 7.03 1 .630084 | 4.03 .871673 7.05 
2 .615619 4.13 .846749 7.02 2 .680326 | 4.05 .872096 7.05 
3 .615867 4.13 . 847170 703 3 .630569 | 4.03 .872519 7.05 
4 .616115 4.12 . 847592 7.03 4 .680811 | 4.05 .872942 COL 
5 .616362 4.13 . 848014 7.03 5 .681054 | 4.038 -878366 | 7.05 
6 .616610 4.12 . 848436 7.03 6 .6381296 | 4.03 . 873789 7.05 
v4 .616857 4.12 .848858 7.03 K. .681588 | 4.03 .874212 7.07 
8 .617104 4.12 .849280 7.08 8 .6381780 | 4.03 .874636 7.05 
9 .617851 4.18 . 849702 7.03 9 .632022 | 4.03 .875059 7.05 
10 .617599 4.10 -850124 7.03 |} 10 .682264 | 4.02 875482 7.07 
11 | 9.617845 4.12 | 9.850546 7.038 || 11 | 9.682505 | 4.08 | 9.875906 7.05 
12 .618092 412 . 850968 203 Mine .682747 | 4.03 .876829 7.05 
13 .618339 4.12 .851390 Lei BY | mas} .6382989 | 4.02 . 876752 7.07 
14 . 618586 4.12 .851812 PoasU > thihies bet .633230 | 4.03 .877176 105) 
15 .618883 4.10 . 852234. 7.08 || 15 .633472 | 4.02 .877599 7.07 
16 .619079 4.12 .852656 7.03 || 16 .683713 | 4.02 .878023 7.05 
ff .619826 | 4.10 . 858078 Fe USS iil a We .683954 | 4.038 .878446 | 7.07 
18 .619572 4.10 . 858500 7.05 || 18 .634196 | 4.02 .878870 1 .OF 
19 .619818 4.12 . 853923 7.038 || 19 .634437 | 4.02 . 879294 7.05 
20 . 620065 4.10 .854845 7.03 || 20 .6384678 | 4.02 879717 7.07 
21 | 9.620811 4.10 | 9.854767 7.03 || 21 | 9.634919 | 4.00 | 9.880141 {ure 
2 . 620557 4.10 .855189 %.05: | "22 .635159 | 4.02 .880565 7.05 
23 .620803 | 4.08 .bo0012 |” 7203 1) 28 .6385400 | 4.02 - 880988 7.07 
24 .621048 4.10 . 8560384 7.03 || 24 .635641 | 4.00 .881412 7.07 
25 . 621294 4.10 . 856456 7.03 \| 25 .6385881 | 4.02 . 881836 7.07 
26 .621540 4.10 .856878 | 7.05 || 26 .636122 | 4.00 -882260 7.05 
27 .621786 | 4.08 8573801 7.03 || 27 .636862 | 4.02 . 882683 7.07 
28 .622031 4.08 .857723 7.03 || 28 .686603 | 4.00 .883107 | 7.07 
29 .622276 4.10 . 858145 7.05 || 29 .636843 | 4.00 .8838531 7.07 
30 .622522 4.08 . 858568 7.03 || 30 .687088 | 4.00 .883955 7.07 
31 | 9.622767 | 4.08 | 9.858990 | 7.05 || 31 | 9.637323 | 4.00 | 9.884379 7.07 
2 .623012 4.08 .859413 W208) iil Oe -6875638 | 4.00 .884803 7.07 
33 .628257 4.08 . 859835 7.05 || 33 -687808 | 4.00 .885227°| 7.07 
84. . 623502 4.08 . 860258 7.03 || 34 .688043 | 4.00 -885651 7.07 
35 .628747 | 4.08 .860680 | 7.05 || 35 . 6382838 | 3.98 .886075 7.07 
36 . 623992 4.08 . 861103 7.03 || 36 .688522 | 4.00 .886499 | - 7.0 
37 .624237 4.07 ~ 861525 7.05 || 37 .638762 | 3.98 . 886923 7.07 
38 .624481 4.08 . 861948 7.03 || 38 -689001 | 4.00 .887847 7.08 
39 .624726 4.07 . 862370 7.05 || 39 .689241 | 3.98 Orie \~ 7.07 
40 .624970 | 4.08 -862793 7.03 || 40 -639480 | 3.98 .888196 7.07 
41 | 9.625215 | 4 07 | 9.863215 7.05 || 41 | 9.689719 | 3.98 | 9.888620 | 7.0 
42 .625459 4.07 . 863638 7.05 || 42 .639958 | 3.98 . 889044 7.08 
43 .625703 4.07 . 864061 7.03 || 43 .640197 | 3.98 . 889469 COE 
44 -625947 4.07 .864483 7.05 || 44 .640436 | 3.98 .889893 7.0 
45 .626191 4.07 - 864906 7.05 || 45 .640675 | 3.98 .890317 7.08 
46 .6264385 | 4.07 .865329 7.05 || 46 .640914 | 3.98 .890742 TORU 
7 . 626679 4.07 . 865752 4.03. || 47 -041153 | 3:97 .891166 7.08 
48 . 626923 4.05 .866174 7.05 || 48 .641891 | 3.98 .891591 7.07 
49 . 627166 4.07 .866597 7.05 || 49 .641630 | 3.97 .892015 7.08 
50 .627410 | 4.07 . 867020 7.05 || 50 .641868 | 3.98 | %892440 7.07 
51 | 9.627654 | 4.05 | 9.867443 | 7.05 || 51 | 9.642107 | 3.97 | 9.892864 | 7.08 
52 .627897 4.05 . 867866 7.05 || 52 642345 | 3.97 . 893289 7.08 
53 . 628140 4.07 . 868289 1.0. || 53 . 642583 | 3.98 .8938714 7.07 
54. . 628384 | 4.05 868712 | 7.05 || 54 - 642822 | 3.97 .894138 7.08 
55 .628627 | 4.05 . 869135 (0p || 55 .648060 | 3.97 .894563 7.08 
56 -628870 | 4.05 . 869558 7.05 || 56 -648298 | 3.95 .894988 | 7.07 
57 .629113 | 4.05 . 869981 7.05 || 57 6435385 | 3.97 .895412 7.08 
58 .629356 | 4.038 . 870404 7.05 || 58 .643773 | 3.97 895837 7.08 
59 . 629598 4.05 - 870827 7.05) || 59 .644011 | 3.97 .896262 7.08 
60 | 9.629841 4.05 | 9.871250 7.05 || 60 | 9.644249 | 3.95 | 9.896687 7.08 — 








~J 


ae 


. 


AND EXTERNAL SECANTS 163 
| 
































56° 57° 
oe 

z Vers. >} D, 1h il exisec, | D2". 4 Verse: | D1". |\¢Rix’ sec Do 1". 
0 | 9.644249 3.95 | 9.896687 |. 7.08 0 | 9.658856 | 3.87 | 9.922247 7.12 
1 644486 | 3.97 .897112 | 7.08 1 .658588 | 3.88 | .922674 | 7.18 
2 64472. 8.95 8975387 | 7.08 2 .658821 | 8.87 .928102 | 7.12 
8 644961 3.95 .897962 7.08 3 659058 | 3.88 - 923529 iad P23 
4 645198 > 3.95 .898387 7.08 4 .659286 | 3.87 .923956 ONS 
5 ,645485 | 3.97 .898812 | 7.08 5 .659518 | 3.87 .924384 } 7.12 
6 645673 3.95 899237 7.08 6 .659750 | 3.88 . 924811 7.138 
i §45910 3.95 .899662 7.08 7 .659983 | 3.87 . 925289 (ey P2 
8 646147 3.95 .900087 7.08 8 .660215 | 3.87 . 925666 Wake 
9 646384 8.93 |' .900512 10'- 9 .660447 | 3.87 . 926094 TAZ 
10 646620 8.95 . 9009388 7.08 || 10 .660679 | 8.85 926521 7.18 
11 | 9.646857 | 3.95 | 9.901863 | 7.08 || 11 | 9.660910 | 8.87 | 9.926949 | 7.13 
“12 . 647094 3.93 .901788 4.08 |) 12 .661142 | 3.87 9273877 7.12 
.13 .647330 8.95 .902213 AO s LS .661374 | 3.85 . 927804 7.13 
14 647567 3.93 .902689 7.08 || 14 .661605 | 3.87 , 928232 7.13 
15 .647803 8.93 . 903064. P10 Web .6618387 | 3.85 . 928660 VAS 
16 .648039 |. 8.95 . 903490 7.08 || 16 .662068 | 8.87 . 929088 Walp: 
17 .648276 8.93 .903915 GAO LE .662800 | 8.85 . 929516 7.13 
18 .648512 38.93 .904841 7.08 || 18 .662531 | 3.85 . 929944 7 A138 
19 .648748 8.98 . 904766 7.10 || 19 .662762 | 8.85 . 930372 7.13 
20 .648984 | 8.93 | .905192 | '7.08 || 20 .662993 | 8.85 .930800 | 7.13 
21 | 9.649220 | 3.93 | 9.905617 | 7.10 || 21 | 9.663224 | 3.85 | 9.931228 | 7.138 
22 . 649456 8.92 . 906043 % 10 || 22 .663455 | 8.85 . 931656 Wetb 
23 .649691 8.93 . 906469 7.08 || 23 .663686 | 8.85 . 982085 7.138 
24 .649927 8.98 - 906894 7.10 || 24 .663917 | 8.85 . 932513 7.18 
25. .650163 8.92 . 907320 FeO. li) 25 .664148 | 3.83 . 9382941 W218 
26 . 650398 8.92 .907746 7.10 || 26 .664878 | 8.85 . 983369 WAB 
27 . 650633 8.98 908172 SLO) [ee .664609 | 3.83 .983798 7 A8 
28 .650869 38.92 .908598 %.10, || 28 .664839 | 8.85 .934226 W 15 
29 .651104 8.92 . 909024 7.10 || 29 .665070 | 8.83 . 984655 7.13 
_ -80 .651889 | 38.92 .909450 | 7.10 || 380 .665300 | 3.83 .9350838 | 7.18 
831 | 9.651574 | 3.92 | 9.909876 | 7.10 || 831 | 9.665530 | 8.88 | 9.985512 | 7.15 
32 .651809 3.92 . 910302 7.10 || 32 .665760 | 8.83 . 935941 fis: 
33 .652044 3.92 -910728 410) 14°33 .665990 | 3.88 . 936369 Wet 
34 .652279 3.92 .911154 7.10 || 34 .666220 | 3.88 . 986798 Wea 
35 .652514 3.90 .911580 7.10 || 35 .666450 | 3.83 . 937227 KEXDS 
36 .652748 3.92 .912006 7.10 || 36 .666680 | 3.838 . 937656 anes 
37 .652983 3.90 . 912482 7.12) 1) 87 .666910 | 8.82 . 988085 {ices 
38 653217 3.92 . 912859 710 188 .667139 | 3.83 .988513 Velo 
39 . 6538452 3.90 . 9138285 7.10 || 389 .667369 | 3.83 . 988942 vem ts) 
40 .653886 | 8.90 | .918711 | 7.12 || 40 .667599 | 3.82 .9893871 | 7.17 
41 | 9.653920 | 3.92 | 9.914138 | 7.10 || 41 | 9.667828 | 3.82 | 9.939801 | 7.15 
42 654155 3.90 .914564 7.12 |) 42 .668057 | 3.88 . 940230 7.15 
43 .654389 3.90 .914991 7.10 || 43 .668287 | 3.82 . 940659 gan is) 
44 . 654623 3.90 .915417 7.12 || 44 .668516 | 3.82 . 941088 elo 
45 . 654857 3.88 .915844 7.10 || 45 .668745 | 8.82 .941517 TELE 
46 .655090 3.90 . 916270 7.12 || 46 ,668974 | 3.82 . 941947 (Eto 
AT .655824 3.90 .916697 7.12 || 47 .669208 | 3.82 . 942376 {6S 
48 6655558 3.90 .917124 7.10 || 48 .669482 | 3.82 942806.) 7.15 
49 . 655792 3.88 . 917550 7.12 || 49 .669661 | 3.80 . 943235 han lee 
50 .656025 3.88 917977 7.12 |) 50 .669889 | 3.82 . 9438665 7.15 
51 | 9.656258 | 3.90 | 9.918404 | 7:12 |] 51 | 9.670118 | 8.82 | 9.944004 | 7.17 
2 . 656492 3.88 .918831 Wels Vivo .670847 | 3.80 .944524 fells: 
53 656725 8.88 | .919258 Gol? |e 58 670575 | 38.82 . 944953 {Osi ff 
54 .656958 8.88 .919685 WAS) Webt .670804 | 3.80 . 945383 felt 
55 657191 3.88 . 920112 12) |) 55 .671082 | 3.80 .945813 ee 
56 657424 3.88 . 920539 Mel Ne 5G .671260 | 3.80 .946243 WV. 
57 .657657 3.88 . 920966 (hob BN atay 9 .671488 | 3.80 .946673 ENG 
58 .657890 | 3.88 . 921393 7.12 || 58 .671716 | 3.82 94.7103 Yea Hf. 
59 . 658123 3.88 . 921820 7.12 || 59 .671945 | 3.78 . 947533 Yi 6 
60 | 9.658856 8.87 | 9.922247 | %.12 || 60 | 9.672172 | 3.80 9.947963 TORT: 


ee 


164 TABLE XXVI.—LOGARITHMIC VERSED SINES 



































58° 59° 
4 Vers. | D..1’, | Ex. sec. | D. 1’. / Vers. , 1, Ex. sec, >. se 
0 | 9.672172 3.80 | 9.947963 7.17 || 0 | 9.685708 | 3.72 | 9.978868 Yo 
1 . 672400 3.80 . 948393 ELT ‘t .685931 | 3.72 . 974302 1.20 
2 . 672628 3.80 . 948823 eG 2) .686154 | 3.72 . 974736 7.22 
3 .672856 3.78 . 9492538 fl 16 3 .686377 | 3.72 .975169 Gros 
4 .673083 3.80 . 949683 7.18 4 .686600 | 3.72 . 975603 703 
5 .673311 3.78 . 950114 eG, 5 .686823 | 3.72 . 976037 "23 
6 6735388 3.80 . 950544. 7.18 6 -687046 | 8.72 39647) 2728 
iG .673766 3.78 . 950975 CEVC ve .687269 | 3.72 . 976905 aos 
8 .673993 3.78 . 951405 7.18 8 .687492 | 3.70 . 977339 ae 
9 . 674220 3.80 . 951836 (ale 9 -687714 | 3.72 907773 % 23 
10 674448 3.78 . 952266 7.18 || 10 .6879387 | 3.70 - 978207 7.23 
11 | 9.674675 3.78 | 9.952697 7.18 || 11 | 9.688159 | 3.72 . 978641 03 
18) .674902 3. %8 . 953128 (ade Mie S .688382 | 3.70 .979075 7.25 
13 .675129 3.78 . 958558 tS Weds .688604 | 3.70 . 979510 7.23 
14 . 675356 Sit . 953989 7.18 || 14 .688826 | 8.70 . 979944 4.25 
15 . 675582 3.78 . 954420 7.18 15 .689048 | 3.72 . 980379 Wiaoes 
16 .675809 3.78 . 954851 7.18 || 16 .689271 | 3.70 . 980813 7.25 
17 . 676036 Stl. . 955282 TAS de, .689493 | 3.70 . 981248 eee 
18 .676262 3.78 .955713 7.18 18 -689715 | 3.70 . 981682 7 25 
19 .676489 Bey . 956144 GAB 19 .689937 | 3.68 . 982117 7 25 
20 . 676715 3.77 . 956575 7.18 || 20 .690158 | 8.70 . 982552 V 25 
21 | 9.676941 | 3.78 | 9.957006 | 7.20 || 21 | 9.690380 | 3.70 | 9.982987 | 7.25 
22 .677168 38.77 . 957488 ff AES 2 .690602 | 3.68 . 9838422 (225 
23 677394 3.7 . 957869 (Colbevallle PAs -690823 | 3.70 . 988857 F225 
24 . 677620 8.00 . 958300 7 20 || 24 -691045 | 3.68 . 984292 % 25 
pas) . 677846 3.77 . 958732 (eS ep: .691266 | 3.70 - 984727 7 25 
26 . 678072 3.77 . 9591638 7.20 || 26 .691488 | 3.68 - 985162 4 25 
27 . 678298 3.75 . 959595 Gavel er .691709 | 3.68 . 985597 42% 
28 . 678523 eG: . 960026 1.20) |) 28 .691930 | 3.68 - 986033 7.25 
29 678749 3.77 . 960458 7.20 || 29 .692151 | 3.68 . 986468 (i227 
30 - 678975 3.%5 - 960890 7.18 || 30 .692372 | 3.68 . 986904 7.25 
31 | 9.679200 | 3.77 | 9.961321 | 7.20 || 31 | 9.692593 | 3.68 | 9.987339 | 7.27 
32 - 679426 3.7) . 961753 p20) Nee 692814 | 3.68 987775 7.25 
33 . 679651 3.75 . 962185 fe UI Ss} -693035 | 3.68 . 988210 here 
34 .679876 afi . 962617 7220) | 34 . 693256 | 3.68 . 988646 Ger! 
35 . 680102 Onta . 963049 7.20 || 85 6938477 | 3.67 - 989082 iets 
36 .680327 3.75 . 963481 7.20 || 36 .693697 | 3.68 .989518 ik 
37 . 680552 3.75 . 963913 {UA B¥e .693918 | 3.67 . 989954 26 
38 680777 3.75 . 964345 7.22 || 38 .694188 | 3.68 .990390 hee 
39 . 681002 38.75 . 964778 7220 W389 .694359 | 3.67 . 990826 A226 
40 . 681227 3.73 . 965210 7.20 || 40 .694579 | 38.67 . 991262 (fe 
41 | 9.681451 8.75 | 9.965642 7,22 || 41 | 9.694799 | 8.67 .991698 terre 
42 .681676 3.75 . 966075 7.20 |; 42 .695019 | 3.68 . 992134 7.28 
43 .681901 3.73 . 966507 7.22 || 43 695240 | 8.67 99257 C26 
44 - 682125 3.75 . 966940 7.20 || 44 .695460 | 3.67 . 993007 7.28 
45 . 682350 3.73 . 967372 1:22 1) 45 .695680 | 3.65 993444 ee 
46 . 682574 3.73 . 967805 7.22 || 46 .695899 | 3.67 . 993880 7.28 
ff . 682798 3), (65 . 968238 7.20 || 47 .696119 | 3.67 . 994817 7 28 
48 .683023 | 3.73 . 968670 | 7.22 || 48 .696339 | 3.67 .994754 | 7.28 
49 . 683247 3.73 . 969103 7.22 || 49 .696559 | 8.65 . 995191 hk 
50 | .683471 | 3.73 -9695386 | 7.22 || 50] .696778 | 3.67 .995627 | 7.28 
51 | 9.683695 | 3.73 | 9.969969 | 7.22 || 51 | 9.696998 | 3.65 | 9.996064 | 7.28 
52 . 683919 3.73 . 970402 7.22 |) 52 .697217 | 3.67 . 996501 7.28 
53 . 684143 3.73 . 970835 7.22 ||| 53 .697437 | 3.65 . 996938 7.30 
54 . 684367 3.72 . 971268 7.22 |) 54 .697656 | 38.65 . 997376 7.28 
55 .684590 ont .971701 W223 |) 55 .697875 | 3.65 . 997813 7.28 
56 . 684814 3.72 . 972135 7.22 || 56 .698094 | 3.65 . 998250 7.28 
57 . 685037 3.73 . 972568 pee Wie7 .698313 | 3.65 . 998687 7.30 
58 .685261 3.72 . 973001 7.23 || 58 .6985382 | 3.65 . 999125 7.28 
59 .685484 3.73 . 973435 7.22 || 59 .698751 | 3.65 | 9.999562 7.30 
60 | 9.685708 3.72 | 9.973868 7:23 |} 60 | 9.698970 | 3.63 110.000000 (30% 
Pt 




















AND EXTERNAL SECANTS TE ELOD 























60° 61° 
’ Vers D. 1”. | Ex. sec. | D. 1’ / Vers. |D.1”".| Ex. sec. |D.1’. 
0 | 9.698970 8.65 | 10.00000: ©,00 0 | 9.711968 | 3.57 | 10.026397 | 7.387 
1 .699189 S203 .000488 | 7.28 1 712182 | 3.58 .026889 | 7.387 
2 .699407 3.65 .00087 7.30 Q 7123897 | 3.57 .027281 | 7.38 
3 - 699626 3.65 .0013813 | 7.380 3 112611 | 3.57 027724 | 7.88 
4 .699845 | 3.63 .001751 | 7.380 4 ~ (12825 .028167 | 7.37 
5 - 700063 3.65 .002189 |; 7.30 5 .713039 .028609 | 7.38 
6 700282 3.63 .002627 | 7.80 6 718253 .029052 | 7.38 
7 .700500 3.63 .008065 | 7.30 v¢ (13467 .029495 | 7.38 
8 . 700718 3.63 .008503 | 7.32 8 . 713681 .029988 | 7.38 
9 . 700936 3.63 .003942 | 7.30 9 713895 .030381 | 7.40 
10 .701154 3.63 .004380 | 7.30 || 10 714109 .0380825 | 4.38 
1i | 9.701372 3:63 | 10.004818 | 7.32 || 11 | 9.414323 10.031268 38 


12 | .701590 | 3.68 .005257 | 7.380 || 12 | .714536 .031711 40 
13 | .701808 | 3.63 .005695 | 7.32 || 18 | .714750 032155 38 


~ 14) .702026 | 3.63 .0061384 | 7.32 || 14 .714963 032598 40 
15 | .702244 | 3.63 .006573 | 7.82 || 15 | .715177 038042 


NN AKAAVIAARIAQGAT GRRRNAVWN 
AP AF ABBA IYI 
tN 
co) 


OVOTENT OLOTOT ON OVOTOUTOTOTVOUOTOTONOU OvVOr Or oy OT Or Or 























16.| .702462 | 3.62 .007012 | 7.30 || 16) .715390 083486 38 
17 | .702679 | 3.63 .007450 | 7.382 || 17 | .715608 -033929 40 
18 | .702897 | 3.62 007889 | 7.32 || 18 | .715817 -034373 40 
19 | .703114 | 3.68 .008328 | 7.382 || 19} .716030 -034817 40 
20 | .703832 | 3.62 .008767 | 7.383 || 20 | .716243 035261 40 
21 | 9.'703549 | 3.62 | 10.009207 | 7.32 || 21 | 9.716456 10.085705 | 7.42 
22 | .703766 | 3.62 .009646 | 7.82 || 22 | .716669 .036150 | 7.40 
23 | .703983 | 38.62 .010085 | 7.33 || 23 | .716882 5 036594 | 7.40 
24 | .704200 | 3.62 010525 | 7.82 |) 24 | .717095 3 037038 | 7.42 
25 | .704417 | 3.62 .010964 | 7.33 || 25 | .7173807 D 037483 | 7.42 
26 | .704634 | 3.62 -011404 | 7.82 || 26 | .717520 3 037928 | 7.40 
27 | .704851 | 3.62 .011843 | 7.88 || 27 | .71'7782 5 038372 | 7.42 
28 | .705068 | 3.62 012283 | 7.33 || 28} .717945 | 3.53 .038817 | 7.42 
29 | .705285 | 3.60 012723 | 7.33 || 29 | .718157 | 3.55 039262 | 7.42 
380 | .705501 | 3.62 .013163 | 7.33 || 80 | .718870 | 3.53 039707 | 7.42 
81 | 9.705718 | 3.62 | 10.013603 | 7.83 || 31 | 9.718582 | 3.53 | 10.040152 | 7.42 
32 | .705935 | 3.60 .014043 | 7.33 || 82 | .718794 | 3.55 040597 | 7.42 
83 | .706151 | 3.60 .014483 | 7.33 || 83 | .719007 | 3.53 041042 | 7.43 
34 | .706867 | 3.62 014923 | 7.33 || 84 | .719219 | 3.53 .041488 | 7.42 
35 | .706584 | 3.60 .015363 | 7.85 || 35 | .719431 | 3.53 .041933 | 7.43 
36 | .706800 | 3.60 .015804 | 7.33 || 36 | .719643 | 3.53 042379 | 7.42 
37 | .707016 | 3.60 -016244 | 7.33 || 87 | .719855 | 3.52 042824 | 7.43 
38 | .707232 | 3.60 .016684 | 7.33 || 88 | .720066 | 3.53 043270 | 7.43 
39 | .707448 | 3.60 -017125 | 7.85 || 89 | .720278 | 3.53 043716 | 7.43 
40 | .707664] 3.60 .017566 | 7.385 || 40 | = .720490 | 38.52 .044162 | 7.43 
41 | 9.707880 | 3.60 | 10.018007 | 7.83 || 41 | 9.720701 | 3.53 | 10.044608 | 7.48 
42 | .708096 | 3.58 -018447 | 7.35 || 42 | .720918 | 3.52 045054 | 7.48 
43 | .708311 | 3.60 .018888 | 7.35 |} 43 | .721124 | 3.53 .045500 | 7.43 
44} .708527 | 3.60 -019329 | 7.35 || 44 | .7213886 | 3.52 .045946 | 7.45 
45 | .708743 | 3.58 .019770 | 7.87 || 45 | .721547 | 3.52 .046393 | 7.43 
46 | .708958 | 3.60 .020212 | 7.85 || 46 | .%21758 | 3.53 046839 | 7.45 
47 | .709174 | 3.58 .020653 | 7.35 || 47 | .721970 | 3.52 047286 | 7.48 
48} .7093889 | 3.58 .021094 | 7.35 || 48] .722181 | 3.52 047732 | 7.45 
49 | .709604 | 3.58 .021585 | 7.37 || 49 | .722892 | 3.52 .048179 | 7.45 
50 | .709819 | 3.58 021977 | 7.37 || 50] 722603 ; 8.52 048626 | 7.45 
51 | 9.710085 | 3.58 | 10.022419 | 7.85 || 51 | 9.722814 | 3.50 | 10.049073 | 7.45 
52 | .710250 | 3.58 022860 | 7.37 || 52 | .728024 | 3.52 049520 | 7.45 
53 | .710465 | 3.58 023302 | 7.37 || 53 |  .728285 | 3.52 .049967 | 7.45 
54.| .710680 } 3.58 023744 | 7.387 || 54 | .723446 | 3.52 050414 | 7.45 
55 | .710895 | 3.57 024186 | 7.387 || 55 | .723657 | 3.50 .050861 | 7.47 
56} .711109 | 3.58 024628 | 7.37 || 56 . (23867 | 3.52 .051309 | 7.45 
57 | .711824 | 3.58 025070 | 7.37 || 57 | .724078 | 3.50 .051756 | 7.47 
58 | .711539 |. 3.57 .025512 | 7.37 || 58 | .724288 | 3.50 052204 | 7.47 
59 | .711753 | 38.58 025954 | 7.38 || 59 | .724498 | 3.52 .052652 | 7.45 
60 | 9.711968 |. 3.57 | 10.026397 | 7.37 || 60 | 9.724709 | 3.50 | 10.053099 | 7.47 
a 


166 TABLE XXVI.—LOGARITHMIC VERSED SINES 


















































62° 63° 
° Vers. | Del’.. |" Eex-séc, (iD. 1".||" “4! Vers: DAI". | Bix See: ay, 123 
0 | 9.724709 3.50 | 10.053099 | 7.47 0 | 9.737200 | 8.43 | 10.080153 | 7.58 
uf 724919 SU Onso4’ | 7.47 1 . 737406 | 8.43 .080608 | 7.57 
2 725129 3.50 .053995 | 7.47 2 .7387612 | 3.48 .081062 | 7.57 
S 725839 3.50 .054443 | 7.48 3 . 7387818 | 3.43 081516 | 7.58 
4 725549 3.50 .054892 | 7.47 4 . 728024 | 3.43 081971 | 7.57 
5 725759 3.50 .0553840 | 7.47 5 . 738236 | 8.48 .082425 | 7.58 
6 725969 3.50 .055788 | 7.48 6 . 7364386 | 3.43 .082880 | 7.58 
ve 726179 3.48 pUGG200 \ita4 cunt, . (88642 | 3.42 .083385 | 7.58 
8 . 7263888 3.50 .056685 | 7.48 8 . 738847 | 3.43 -083790 | 7.58 
9 . 726598 38.50 .057184 | 7.48 9 739053 | 3.42 .084245 | 7.58 
10 726808 3.48 .057583 | 7.48 || 10 - 789258 | 3.43 .084700 | 7.58 
WEN AD evrew dla lrg 3.50 | 10.058032 | 7.48 || 11 | 9.789464 | 3.42 | 10.085155 7-60 
12 Metal 3.48 .058481 | 7.48 12 . 739669 | 8.43 .085611 | 7.58 
13 127436 3.48 .058930 | 7.48 || 13 . 7389875 | 8.42 .086066 | 7.60 
14 . 727645 38.50 .059379 | 7.48 || 14 . 740080 | 8.42 -086522 | 7.58 
15 727855 3.48 .059828 | 7.50 || 15 - 740285 | 3.42 .086977 | 7.60 
16 728064 3.48 .060278 | 7.48 || 16 . 740490 | 8.42 087433 | 7.60 
17 (28278 3.48 SOOO 24 (heme ela: 140695 | 3.42 - 087889 | 7.60 
18 . (28482 3.48 061177 | 7.48 18 . 740900 | 3.42 .088345 | 7.60 
19 . 728691 3.48 .061626 | 7.50 || 19 .741105 | 8,42 .O88801 | 7.62 
20 . 728900 3.48 .062076 | 7.50 || 20 . 7413810 | 8.42 -089258 | 7.60 
21 | 9.729109 3.47 | 10.062526 | 7.50 || 21 | 9.741515 | 3.40 10.089714 | 7.62 
22 UBL 3.48 .062976 | 7.50 22 (41719 | 8.42 .090171 | 7.60 
23 129526 3.48 .0638426 | 7.50 23 . 741924 | 3.42 090627 | 7.62 
24 (29735 3.47 .068876 | 7.52 || 24 - 742129 | 3.40 -091084 | 7.62 
25 . 729943 3.48 .064827 | 7.50 || 25 . 142333 | 8.42 .091541 | 7.62 
26 . 7380152 rately 064777 | 7.50 26 . 742588 | 3.40 .091998 | 7.62 
i . 780860 3.48 065227 | 7.52 || 27 . 142742 |. 8.40 .092455 | 7.62 
28 . (380569 3.47 .065678 | 7.52 || 28 . 742946 | 8.40 092912 | 7.63 
29 130777 3.47 .066129 ; 7.52 || 29 . 743150 | 8.42 .093870 | 7.62 
30 . 730985 3.47 .066580 | 7.50 || 80 . 748355 | 8.40 -098827 | 7.68 
31 | 9.731193 3.47 | 10.067030 | 7.53 || 31 | 9.743559 | 8.40 | 10. 094285 | 7.63 
82 . 731401 3.47 067482 | 7.52 V4 (49762 | 8.40 .094743 | 7.62 
33 . 731609 3.47 .067933 | 7.52 33 . 748967 | 3.40 -095200 | 7.63 
34 Bi foiiRoil bi 3.47 .068384 | 7.52 34 744171 | 8.40 .095658 | 7.63 
385 . 732025 3.47 .068835 | 7.53 || 85 - 744875 | 3.38 096116 | 7.65 
36 Py 2 423% § 3.47 .069287 | 7.52 || 36 74457 3.40 -096575 | 7.63 
37 (32441 3.45 -069738 | 7.53 37 . 744782 | 3.40 -097033 | 7.63 
38 . 732648 3.47 .070190 | 7.53 38 . 744986 | 3.38 -097491 | 7.65 
39 «732856 8.47 -070642 | 7.52 39 . 745189 | 3.40 .097950 | 7.63 
40 . 733064 3.45 .071093 | 7.53 || 40 - 745393 | 3.38 -098408 | 7.65 
41 | 9.738271 8.45 | 10.071545 | 7.55 | 41 | 9.745596 | 8.40 | 10.098867 | 7.65 
42 . 138478 8.47 .071998 | 7.53 || 42 . 745800 | 3.38 -099326 | 7.65 
43 . 733686 8.45 .072450 | 7.53 || 43 . 746008 | 3.38 .099785 | 7.65 
44 7338893 83.45 .072902 | 7.53 44 . 746206 | 3.38 .100244 | 7 67 
45 . 734100 8.45 .073354 | 7.55 || 45 746409 | 3.40 .100704 | 7.65 
46 . 734307 B.47 .073807 | 7.55 || 46 . 746618 | 8.38 -101163 | 7.67 
47 «7384515 8.43 .074260 | 7.53 47 . 746816 | 3.38 -101623 | 7.65 
48 . 734721 8.45 074712 | 7.55 || 48 - 747019 | 3.38 - 102082 | 7.67 
49 . 7384928 8.45 .075165 | 7.55 49 Pal Pf Pade lanes ery . 102542 | 7.67 
50 . 735135 8.45 .075618 | 7.55 || 50 . 747424 | 3.38 .103002 | 7.67 
51 | 9.735342 _ 8.45 | 10.076071 | 7.55 || 51 | 9.747627 | 3.38 10.103462 | 7.67 
52 735549 8.43 076524 | 7.55 || 52 . 747830 | 3.38 - 103922 | 7.67 
53 . (00155 3.45 O76977 | 7.57 || 53 . 748033 | 3.87 .104882 | 7.68 
54 - 735962 3.45 077431 | 7.55 || 54 . 748235 | 3.388 .104843 | 7.67 
55 . 736169 3.43 Ovary Woe ||. 55 . 748488 | 3.37 .105308 | 7.68 
56 7363875 3.43 078388 | 7.57 || 56 . 748640 | 3.38 - 105764 | 7.67 
57 . 736581 3.45 078792 | 7.55 || 57 748843 | 3.37 . 106224 | 7.68 
58 (36788 3.43 2079245 | 7.57 || 58 .749045 | 3.37 .106685 | 7.68 
59 . (86994 3.43 .079699 | 7.57 || 59 749247 | 3.37 .107146 | 7.68 
60 | 9.737200 3.43 | 10.080153 | 7.58 || 60 | 9.749449 | 3.38 10.107607 | 7.70 - 


AND EXTERNAL SECANTS HOT 








64° 65° 







































































4 Vers. |D. 1’.| Ex. sec. |D 1’. f Vers. |D.1".| Ex. sec. |D.1". 
0 | 9.749449 | 3.388 | 10.107607 7.70 0 | 9.761463 | 3.30 | 10.185515 | 7.82 
1 .749652 | 3.37 .108069 | 7.68 1 .761661 | 3.382 .135984 | 7.83 
2) .749854 | 3.37 .108530 | 7.70 2 .761860 | 3.30 .136454 | 7.82 
3 .750056 | 3.37 . 108992 7.68 3 .762058 | 3.30 .1386923 | 7.83 
4 .750258 | 3.35 . 109453 7.70 4 . 762256 | 3.30 .137393 | 7.83 
5 | .%u459 |° 3:37 .109915 | 7.70 5 162454 | 3.30 .137863 | 7.83 
6 .700661 | 3.37 SHOST |) Ven UNO .762652 | 3.30 . 138333 | 7.83 
7 | .%o0u863 | 3.37 .1108389 | 7.7 Ff 762850: | 3.28 .138803 | 7.83 
8 .751U65 | 3.35 .111301 WOME 8 .763047 | 3.30 .189273 | 7.85 
9 .751266 | 3.37 .111763 fh fh 9 . 763245 | 3.30 .139744 | 7.83 
10 .751468 | 3.35 a i ode Teele 10 . 763443 | 3.30 .140214 | 7.85 
141.| 9.751669 | 3.37 | 10.112688 | 7.72 || 11 | 9.763641 | 3.28 | 10.140685 | 7.85 
43 .751871 | 3.35 .113151 ate 12 .763888 | 3.30 .141156 | 7.85 
13 (52002 | O.0D .113614 (eae 13 .764036 | 3.28 .141627 | 7.85 
44 SERA2TS |) 8380 SUT407T Ee Fone 764283 | 3.28 .142098 | 7.85 
15 752475 | 3.35 .114540 (He? 15 .764480 | 38.30 .142569 | 7.87 
16 152676 | 3.35 .115003 (ON? 16 . 764628 | 3.28 .143041 | 7.85 
17 DST) otoe .115466 fay ur .764825 | 3.28 .143512 | 7.87 
18 . 758078 | 3.35 .115929 aie 18 .765022 | 3.28 . 143984 | 7. 
19 .7538279 | 3.35 .116393 Toto 19 .765219 | 3.28 .144456 | 7.87 
20 . 753480 | 3.35 .116857 13 20 .765416 | 3.28 .144928 | 7.87 
91 | 9.753681 | 3.33 | 10.1173821 | 7.7 21 | 9.765613 | 3.28 | 10.145400 | 7.87 
22 .753881 | 3.35 Livan ete colice 765810 | 3.28 .145872 | 7.88 
93 . 754082 -| 3.35 . 118249 Fai Nees: .766007 | 3.28 .146345 | 7.88 
24 . 754283 | 3.33 .118713 Gate 24 766204 | 3.28 .146818 | 7.87 
25 .754483 | 3.35 119177 | %.%5 || 25 .766401 | 8.27 .147290 | 7.88 
26 . 754684 | 3.33 .119642 T7383. Wi 26 . 766597 | 3.28 .147763 | 7.88 
Q7 754884 | 3.385 . 120106 oat Niner . 766794 | 8.28 .148236 | 7.90 
28 755085 | 3.33 pili brah ape yeal 28 .766991 | 3.27 .148710 | 7.88 
29 . (55285 | 3.33 .121036 "UD | 29 .767187 | 3.28 .149183 | 7.90 
380 | .755485: | 3.33 .121501 | 7.75 || 30 767384 | 3.27 .149657 | 7.88 
31 | 9.755685 | 3.35 | 10.121966 | 7.75 || 31 | 9.767580 | 3.27 | 10.150130 | 7.90 
32 . 755886 | 3.33 .122431 ek 82 767776 | 8.27 .150604_| 7.90 
33 .756086 | 3.33 . 122897 7.%5 || 33 767972 | 3.28 .151078 | 7.90 
34 .756286 | 3.33 . 123862 WNT || 34 .768169 | 3.27 .151552 | 7.92 
35 .756486 | 38.32 . 123828 CEA it, . 768365 | 3.27 .152027 | 7.90 
36 .756685 | 3.33 124294 Te aicou .768561 | 3.27 .152501 | 7.92 
37 -756885 | 3.33 . 124760 W770 || 387 168757 | 3.27 .152976 | 7.90 
38 .757085. | 3.33 . 125226 TUT \! 388 .768953 | 3.27 .153450 | 7.92 
39 757285 | 3.32 . 125692 ZV |) 39 . 769149 | 3.25 .153925_| 7.92 
40 | .757484 | 3.33 .126158 | 7.78 || 40 769844 | 38.27 .154400 | 7.98 
4i | 9.457684 | 3.32 | 10.126625 | 7.78 || 41 | 9.769540 | 3.27 | 10.154876 | 7.92 
42 .7578838 | 3.33 .127092 alg 42 .769736 | 3.25 .155351 | 7.92 
43 2758083 | 3.32 127558 7.78 || 43 .769981 | 3.27 .155826 | 7.98 
44 . 758282 | 3,32 .128025 okt 44 100127 | 3.27 .156302 | 7.938 
45 . 758481 | 3.33 . 128492 7.80 || 45 770823: | 3.25 .15677 7.938 
46 .758681 | 3.32 . 128960 7.8 || 46 .770518 | 3.25 157254 | 7.93 
47 758880 | 3.32 .129427 7.78 || 47 7707138 | 38.27 .157730 | 7.93 
48 75907 3.382 . 129894 7.80 || 48 .770909 | 3.25 .158206 | 7.95 
49 .759278 | sy] . 130362 7.80 || 49 .771104 | 3.25 .1586838 | 7.93 
50 .759477 | 3.32 .130830 | 7.80 || 50 771299 | 3.25 .159159 | 7.95 
51 | 9.759676 | 3.32 | 10.131298 | 7.80 || 51) 9.771494 | 3.25 | 10.159636 | 7.95 
52 .75y875 | 3.30 . 1381766 7.80 || 58 .771689 | 38.25 .160113 | 7.95 
53 .760U73 | 3.32 1382234 | 7.80 || 53 ~771884 | 3.25 .160590 | 7.95 
54 .W6uz72 | 3.3824. 182702 7.80 || 54 .772079 | 3.25 .161067 | 7.97 
55 .76U471 | 3.380 . 183170 7.82 || 55 772274 | 3.25 .161545 | 7.95 
56 .760669 | 3.82 . 188639 7.82 || 56 .7'72469 | 3.25 .162022 | 7.97 
57 .760868 | 3.30 . 184108 7.82 || 57 .772664 | 3.23 .162500 | 7.97 
58 .761066 | 3.32 . 134577 7.82 || 58 772858 | 3.25 .162978 | 7.97 
59 .761265 | 3.30 . 135046 7.82 || 59 «7738058: | 3.25 .163456 | 7.97 
60 | 9.761463 | 3.30 | 10.135515 | 7.82 || 60 | 9.778248 | 3.23 | 10.163934 | 7.98 


4 
, 


eT | 


168 TABLE XXVI.—LOGARITHMIC VERSED SINES 
V—<—<——_—_—_—nkeeeeeee ee _c=c_ 


| 





, Vers. 
0 | 9.773248 
1 773442 
2 773636 
3 773831 
4 774025 
5 774219 
6 . 774414 
7 . 774608 
8 . 774802 
9 - 774996 
10 - 775190 
11 | 9.775384 
12 Sy ithaay ai 
13 YW 
14 775965 
15 776159 
16 776352 
Le 776546 
18 (76739 
19 - 776933 
20 777126 
21 | 9.777319 
22 777512 
5 - 777705 
24 . 777899 
25 . 778092 
26 778285 
27 -VI8477 
28 . 778670 
29 - 778863 
30 779056 





31 | 9.779248 





32 | .779441 
33 | .7796384 
34} .779826 
35 | .780018 
36 | .780211 
37 | .780403 
88 | .780595 
39 | .780787 
40 | .780980 
41 | 9.781172 
42 | .781364 
43 | .781556 
44} .781747 
45 | .781939 
46 | .782131 
47 | .782823 
48 | .782514 
49 | .782706 
50 | .782897 
51 | 9.783089 
52 | .783280 
53 | .783471 
54 | .783663 
55 | .788854 
56 | .784045 
57 |  .784236 
58 | . 784427 
59 | .784618 
60 ! 9.784809 











66° 


Dae 


3.23 








3.18 
3.18 


Ex. sec, 


10.163934 
-164413 
- 164891 
165370 
. 165849 
166328 
166807 
167286 
167766 
168245 
168725 

10. 169205 
. 169685 
170165 
170646 
171127 
-171607 
172088 
172569 
173051 
173532 


10.1%4014 
174496 
174978 
- 175460 
175942 
176425 
176907 
177390 
177878 
178356 


10. 178839 
179823 


"183197 


10. 183682 
. 184167 
. 184653 
185138 
. 185624 
. 186110 
. 186596 
. 187082 
. 187568 
. 188055 


10. 188542 
189029 
189516 
- 190008 
190491 
-190978 
. 191466 
191954 
192443 

10.192931 





D1". 


Sse 


eooooofo 
“FE OULOTOLOTOO OFT ON COON WW WWWOHWW 


QS 


SS SSS Oe Oo oOo SS SCS So SOS SSooaqoq0q> 


GP 00.00.00 0.00 90.90. Ge.50_GP.GP.GP.G0 0.99 d0.40.00 00 G0 Ge.G0. 60.00.90. 0 50.5 490.90 © OP 26 BP Ge GP G0.G0 _Ge.N0 40 40 40 40 G0 GO O0 « 
KNWMIWWWWWWW WWOOTSOCOHOSG GBHGDRIIOBIA 


Pah ek ek ee ee ek ee 














67 





= ~ 
SOMIODUIRWWOeHO 


Vers. |D.1’.| Ex. sec, |D.1”. 


9.784809 
785000 
(85191 
. 785381 
785572 
185763 
(85953 
786144 
786334 
786524 
(86715 


. 786905 
(87095 
(87285 
«187475 
787665 
187855 
. 788045 
~ (88235 
«188425 
788614 


9.788804 
788993 
489183 
(89372 
189562 
- 189751 
. 789940 
- 790130 
- 790319 
- 790508 


9.'790697 
. 790886 
791075 
791264 
191453 
- 791641 
791830 
792019 
(92207 
- 792396 


9.792584 
792772 
792961 
793149 
~ 193337 
- 793525 
793714 
793902 
794090 
(94277 


9.794465 
794653 
794841 
795028 
795216 
~ 795404 
795591 
795779 
(95966 

9.796153 


Oo 














3.17 


co to 
z 


rea Ss 
or 








10.192931 
193420 
198908 
194897 
194886 
195376 
. 195865 
196355 
196845 
197335 
197825 


10.198315 
. 198806 
199297 
199788 
200279 
200770 
- 201262 
201753 
202245 
202737 


10.208229 
203722 
204215 
204707 
- 205200 
205694 
206187 
206681 
207174 
207668 

10.208162 
208657 
209151 
209646 
210144 
210636 
211131 
211627 
212123 
212618 


10.213115 
218611 
214107 
214604 
215101 
-215598 
-216095 
-216593 
-217090 
217588 


10.218086 
-218585 
-219083 
219582 
220081 
-220580 
-221079 
221578 
222078 

10. 222578 








G2 Gb G0 G0 GO.00 G0 G0 Ge Go G0 GO. G0. G0 G0. 20.00 GD. GD G0.00.G0:00.00 00 00,00 G0 


CVODEDED COED EDOD GOdDIO DD DW Ww wd AW ID WI DDD wrmwwwwnwy» wwwHe 
BELERBRRSSS SSRSPRRRNY PBRVVLVYNYL PYRVPNVLLY Were 


8 
8 
8 
8 
8 
8 
8 
8. 
8 
8 
8 
8 
8 
8 
8 
8 


8. 


AND EXTERNAL SECANTS 169 








68° 69° 
































Vers, Dit Ex.sees- | Dalit)? Versi ga) Doe. |p ix sees Dil. 

0 | 9.796158 | 3.13 | 10.222578 33 0 | 9.807286 YW | 10.252957 | 8.55 
1 .796341 | 3.12 . 2238078 reat (| a | .807470 if 220047 8.55 
2 - 796528 | 3.12 6220578 35 2 .807654 i .2539838 | 8.57 
3 OC15 [3.12 . 224079 33 3 .807837 MG .254497 | 8.55 
4 - 796902 | 8.12 .224579 85 4 . 808021 5 .255010 | 8.57 
5 .797089 | 3.12 . 225080 8D ||. 5 .808204 a .255524 | 8.58 
6 -797276 | 3.12 . 225581 37 || 6 .808388 5D . 256039 | 8.57 
ff (9/463 | 3.12 . 226083 So ein & .808571 if .206558 | 8.58 
8 .797650 | 3.12 . 226584 ot 8 ~ 808755 i) .257068 | 8.57 
9 . 797837 | 3.10 . 227086 oH 1am) .808938 i .257582 | 8.60 
10 - 798023 | 3.12 6227588 eel O . 809121 Va .258098 | 8.58 
11 | 9.798210 | 3.12 | 10.228090 11 | 9.809305 5 | 10.258618 | 8.60 
12 | .798397 | 3.10 - 228592 12 . 809488 5 .259129 | 8.58 
13 .798583 | 3.12 .229095 183 .809671 5 .259644 | 8.60 
14 T9877 3.10 .229598 14 . 809854 I) .260160 | 8.62 
15 .798956 | 3.10 .230101 15 .810037 5 .260677 | 8.60 
16 .799142 | 3.12 . 230604 16 .810220 5 .261193 | 8.62 
17 .799829 | 3.10 201107 ive .8104038 3 .261710 | 8.62 
18 .799515 | 3.10 .231611 18 ~810585 bs) .262227 | 8.62 
19 .799701 | 3.10 202115 19 .810768 D . 262744 | 8.638 
20 799887 | 3.12 .2382619 20 .810951 5 . 263262 | 8.62 


21 | 9.800074 | 8.10 | 10.233128 
22 | .800269 | 3.10 208027 


10.268779 | 8.63 
264297 | 8.63 


21 | 9.811134 
22, .811316 












































8. 3.0 

8. 3.0 

8. 3.0 

8. 3.0 

8. 3.0 

8. 3.0 

8. 3.0 

8. 3.0 

8. 3.0 

8.: 3.0 

8.3 3.0 

8.37 | 3.0 

8.38 3.0 

8.38 3.0 

8.38 3.0 

8.38 | 3.0 

8.38 3.0 

8.40 3.0 

8.40 3.0 

8.40 3.0 

8.40 3.0 

8 40 3.03 

8.42 8.05 
23 | .800446 | 3.08 -204182 | 8.42 || 28 .811499 | 3.03 .264815 | 8.65 
24 | .800631 | 3.10 .204637 | 8.42 || 24] .811681 | 3.05 .265334 | 8.65 
25 | .800817 | 3.10 .205142 | 8.42 || 25 .811864 | 38.03 .265853 | 8.63 
26 | .801003 | 3.10 .280647 | 8.43 || 26 | .812046 | 8.03 .266871 | 8.67 
27 | .801189 | 3.10 .286153 | 8.42 || 27 | .812228 | 3.03 .266891 | 8.65 
28 | .801375 | 3.08 .236658 | 8.43 |} 28 .812410 | 3.05 -267410 | 8.67 
29 | .801560 | 3.10 .23/164 | 8.43 |} 29 812593 | 38.08 -267980 | 8.65 
30 | .801746 | 3.08 | .287670 | 8.45 || 80 | .812775 | 3.03 .268449 | 8.68 
31 | 9.801931 | 3.10 | 10.238177 | 8.43 || 381 | 9.812957 | 3.03 | 10.268970 | 8.67 
32 | .802117 | 3.08 .288683 | 8.45 2 | .813139 | 3.03 .269490 | 8.68 
33 | .802802 | 3.08 .239190 | 8.45 || 83] .813821 | 3.03 .270011 | 8.67 
34 | 802487 | 3.10 .239697 | 8.45 || 384 8135038 | 3.03 .270531 | 8.68 
35 | .802673 | 3.08 .240204 | 8.47 || 35 .818685 | 3.02 -271052 | 8.70 
36 802858 | 3.08 -240712 | 8.45 || 86 .813866 | 3.03 .271574 | 8.68 
37 | .803043 | 3.08 -241219 | 8.47 || 37 .814048 | 38.03 -212095 | 8.70 
38 | .803228 | 3.08 241727 | 8.47 || 88 .814230 | 3.02 .212617 | 8.70 
39 | .803418 | 3.05 242235 | 8.48 || 3: .814411 | 3.03 .273139 | 8.72 
40 | .808598 | 3.08 242744 | 8.47 || 40 | .814593 | 3.03 273662 | 8.70 
41 | 9.803783 | 3.08 | 10.243252 | 8.48 || 41 | 9.814775 | 3.02 | 10.274184 | 8.72 
42 | .803968 | 3.08 243761 | 8.48 || 42] .814956 | 3.02 2014007 | 8.72 
43 | .804153 | 3.08 .244270 | 8.48 || 43 | .815137 | 3.03 .275280 | 8.72 
44 | .804338 | 3.07 2447F 8.50 || 44 .815319 | 38.02 275753 | 8.78 
45 | .804522 | 3.08 .245289 | 8.48 || 45 .815500 | 3.02 R16277 | 8.7 
46 | .804707 | 3.08 -245798 | 8.50 || 46 | .815681 | 3.02 .276801 | 8.73 
47 | .804892 | 3.07 246308 | 8.50 || 47 | .815862 | 3.03 277825 | 8.73 
48 | .805076 | 3.08 .246818 | 8.52 || 48 .816044 | 3.02 277849 | 8.75 
49 | .805261 | 3.07 247329 | 8.50 || 49 | .816225 | 3.02 278874 | 8.7 
50 | .805445 | 3.07 247839 | 8.52 || 50; .816406 | 3.02 .278899 | 8.7 
51 | 9.805629 | 3.08 | 10.248350 | 8.52 || 51 | 9.816587 | 3.00 | 10.279424 | 8.75 
52 | .805814 | 3.07 .248861 | 8.52 || 52] .816767 | 3.02 279949 | 8.7 
53 | .805998 | 3.07 249872 | 8.52 || 53 | .816948 | 3.02 280475 | 8.7 
54} .806182 | 3.07°(* .249883 | 8.53 || 54] .817129 | 3.02 281000 | 8.78 
55 | .806366 | 3.07 .250895 | 8.53 || 55 | .817310 | 3.00 281527 | 8.7 
56 | .806550 | 3.07 .250907 | 8.53 || 56 | .817490 | 3.02 282053 | 8.7 
57 | .806734 | 3.07 -251419 | 8.55 || 57 | .817671 | 8.02 282580 | 8.77 
58 | .806918 | 8.07 .2519382 | 8.53 || 58] .817852 | 3.00 .283106 | 8.80 
59 | .807102 | 3.07 .252444 | 8.55 |) 59] .818032 | 3.02 283634 | 8.78 
60 | 9.807286 | 3.07 | 10.252957 | 8.55 || 60 1 9.818213 |! 3.00 8.80 


10.284161 





170 TABLE XXVI.—LOGARITHMIC VERSED SINES ~ 


an 


















































70° : ic 
! 
, Vers. | D.1".| Ex. sec. | D. 1’. , Vers. |D.1°.| Ex. sec. [p.t. 
O | 9.818213 | 8.00 | 10.284161 8.80 0 | 9.828938 | 2.95 | 10.816296 | 9.07 
1 .8183893 | 3.00 284689 8,78 1 .829115 | 2.95 .816840 | 9.08 
2 .818573 | 3.02 . 285216 8.82 2 .829292 | 2.95 .oL7885 | 9.07 
3 .818754 | 3.00 .285745 8.80 3 .829469 | 2.95 .017929 | 9.10 
4 .818934 | 3.00 .286273 8.82 4 829646 | 2.95 .318475 | 9.08 
5 .819114 | 3.00 . 286802 8.82 5 829823 | 2.95 .319020 | 9.08 
6 .819294 | 3.00 .2873381 8.82 6 830000 | 2.95 019565 | 9.10 
p .819474 | 3.00 .287860 8.82 fs 830177 | 2.93 “S20h11 | 92 
8 .819654 | 3.00 .288389 8.83 8 8380353 | 2.95 .3820658 | 9.10 
9 .819834 | 3.00 . 288919 8.83 9 .830530 | 2.98 .321204 | 9.12 
10 .820014 | 3.00 . 289449 8.83 ||} 10 .8380706 | 2.95 -o2l751 | 9.12 
11 | 9.820194 | 3.00 | 10.289979 8.85 || 11 | 9.880888 | 2.93 | 10.322298 | 9.12 
2 .820874 | 2.98 .290510 8.55 12 .881059 | 2.95 822845 | 9.18 
13 .820553 | 3.00 .291041 8.85 13 . 831236 | 2.93 ,020093 | 9.18 
14 .820733 | 3.00 .291572 8.85 14. .831412 | 2.95 323941 | 9.13 
15 .8209138 | 2.98 . 292108 8.87 || 15 .8381589 | 2.98 3824489 | 9.15 
16 .821092 | 3.00 . 292635 8.85 16 .831765 | 2.93 325088 | 9.15 
17 .821272 | 2.98 . 293166 8.87 17 .831941 | 2.938 825587 | 9.15 
18 .821451 | 3.00 . 293698 8.88 18 . 882117 | 2.98 326136 | 9.17 
19 .821631 | 2.98 . 294931 8.88 19 .882293 | 2.93 326686 | 9.15 
20 .821810 | 2.98 . 294764 8.87 || 20 .832469 | 2.93 8272385 | 9.18 
21 | 9.821989 | 2.98 | 10.295296 8.90 || 21 | 9.882645 | 2.93 | 10.827786 | 9.17 
22 .822168 | 3.00 . 295830 8.88 || 22 , 8382821 | 2.938 ,928836 | 9.18 
33 . 822348 | 2.98 . 296363 8.90 || 23 .8382997 | 2.93 .828887 | 9.18 
24 . 822527 | 2.98 .296897 8.90 || 24 ~8a0l73 | -2.938 .829438 | 9.18 © 
25 .822706 | 2.98 .297431 8.90 || 25 .8388849 | 2.93 .829989 | 9.20 
26 .822885 | 2.98 . 297965 8.92 || 26 .83838525 | 2.92 .300541 | 9.20 
27 823064 | 2.98 | .298500 8.90 || 27 .883700 | 2.93 .331093 | 9.20 
28 , 823243 | 2.97 .299034 8.93 || 28 .883876 | 2.92 ~o51045 | 9,22 
29 ,823421 | 2.98 .299570 8.92 || 29 , 8384051 | 2.93 082198 | 9.20 
30 | .823600 | 2.98 | 800105 | 8.93 |] 80 | -834227 | 2.92] 1882750 | 9.23 
31 | 9.828779 | 2.98 | 10.300641 8.92 || 81 | 9.884402 | 2.93 | 10.883304 | 9.22 
32 .823958 | 2.97 .801176 8.95 || 32 .884578 | 2.92 .d80807 | 9.23 
33 .824136 | 2.98 .301713 8.93 || 33 8384753 | 2.92 .304411 | 9.28 
34 .824315 | 2.97 . 802249 8.95 || 34 .834928 | 2.93 .834965 | 9.25 
35 .824493 | 2.98 .3802786 8.95 || 35 805104 | 2.92 .830520 | 9.23 
36 824672 | 2.97 - 3803323 8.95 || 36 .835279 | 2.92 .3836074 | 9.25 
37 .824850 | 2.97 .3803860 8.97 || 387 835454 | 2.92 .336629 | 9.27 
38 .825028 | 2.98 -804398 8 97 || 38 .835629 | 2.92 .8387185 | 9.27 
39 825207 | 2.97 .804936 8.97 || 39 .835804 | 2.92 807741 | 9.27 
40 .825385 | 2.97 3805474 8.97 || 40 .885979 | 2:92 .8388297 | 9.27 
41 | 9.825563 | 2.97 | 10.306012 8.98 || 41 | 9.836154 | 2.92 | 10.388853 | 9.28 
42 .825741 | 2.97 . .806551 8.98 || 42 8363829 | 2,92 .839410 | 9.28 
43 825919 | 2.97 .807090 8.98 || 43 .836504 | 2.90 .839967 | 9.28 
44 .826097 | 2.97 .807629 9.00 || 44 .886678 | 2.92 .340524 | 9.30 
45 826275 | 2.97 .3808169 8.98 || 45 .886853 | 2.92 .841082 | 9.30 
46 .826453 | 2.97 - 3808708 9.02 || 46 .8387028 | 2.90 .841640 | 9.30 
47 .826631 | 2,97 . 309249 9.00 || 47 .8387202 | 2.88 .042198 | 9.30 
4§ .826809 | 2.97 . 809789 9.02 || 48 .8387377 | 2,90 842756 | 9.32 
49 .826987 | 2.95 .310330 9.02 || 49 .Ba(bol | 2,92 .049315 | 9.33 
50 .827164 | 2,97 .310871 9.02 || 50 .887726 | 2.90 .043875 | 9.382 
51 | 9.827342 | 2,95 | 10.311412 9.02 || 51 | 9.837900 | 2.92 | 10.344434 | 9.33 
52 .827519 | 2,97 .38119538 9.03 || 52 .888075, | 2.90 844994 | 9.33 
53 .827697 | 2.95 -312495 9°03 ||| 538 .8388249 | 2.90 .3845554 | 9.35 
54 827874 | 2.97 .818037 9.05 || 54 . 838428 | 2,90 846115 | 9.35 
55 828052 | 2.95 .313580 9.03 || 55 .838597 | 2.90 .846676 | 9.35 
56 828229 | 2.95 . 314122 9.05 || 56 .838771 | 2.90 .047237 | 9.35 
57 ,828406 | 2.97 .314665 9.07 || 57 -838945 | 2.90 847798 | 9.37 
58 .828584 | 2.95 .815209 9.05 || 58 -839119 | 2.90 .3848360 | 9.37 
59 .828761 | 2.95 .815752 9.07 || 59 .839293 | 2.90 .3848922 | 9.38 
60 | § 828938 | 2.95 | 10.316296 9.07 |} 60 | 9.839467 | 2.90 | 10.849485 | 9.38 
LE 
































382120 
382703 
383286 


72° 
, Vers. |D.1°,| Ex. sec. 
0 | 9.839467 | 2.90 | 10.349485 
1| 839641 | 2/90 | 350048 
2] |839815 | 2.90 | 1350611 
3| 830989 | 2.88 | “351175 
4} ‘840162 |°2.90 | 1351738 
5 | (840336 | 2.90 | 1352803 
6| 840510 | 2.88 | “352867 
7 | |840683 | 2:90 | _353432 
8 | 840857 | 2.88 | 1353997 
9} ‘841030 | 2:90 | 1354568 
10} [841204 | 2.88 | 1355129 
11 | 9.841377 | 2.88 | 10.355695 
12 | |841550 | 2.88 |  _356261 
13) ‘841723 | 2’88 | 356828 
14! '841896 | 2.90} 1357395 
15 | ‘842070. | 2°88 | 1357963 
16} 1842243 | 288 | 1358531 
7 | 1842416 | 2.88} 1359099 
18 | |842589 | 2.88 | 359668 
19} “s4e762 | 2'87 | ‘360237 
20 | (842934 | 2.88]  |360806 
21 | 9.843107 | 2.88 | 10.361376 
22 | 1843280 | 2°88 | 361946 
23 | 943453 | 2/87 | 1362516 
24 |843625 | 2.88 | 1363087 
25 | (843798 | 2.87 | 1363658 
26 | (843970 | 2.88 | 364229 
97 | (844148 | 2187 | 364801 
28 | (844315 | 2.88] 365373 
99 | (8444885 2.87 | 365945 
-30 | 844660 | 2°87 | 1366518 
31 | 9.844832 | 2.87 | 10.367091 
32 | .845004 | 2.88 | ‘367665 
33 | 1845177 | 2.87 | 1368289 
34| 1845349 | 2.87 | 368813 
35 | 1845521 | 2°87 | 369387 
36| 1845693 | 2.87 | “3690962 
37 | 1845865 | 2/87 | '370538 
38} 846087 | 2.85 | '371113 
39 | 846208 | 2.87 | ‘371689 
40 | .846380 | 2.87 | 372266 
41 | 9.846552 | 2.87 | 10.372842 
42 | .846724 | 2°85 | 373419 
43 | 1846895 | 2.87 | _373997 
44) ‘8470671285! “374575 
45 | (847298 | 287 | 1375153 
46 | 1847410 | 2.85 | “375731 
47 | (847581 | 2.87 | ‘av6310 
48 | (847753 | 2.85} ‘376890 
49 | “847924 | 2°85 | _377469 
50, 848095 | 2.87 | ‘378049 
51 | 9.848267 | 2.85 | 10.378630 
52) _848438 | 2.85 379210 
53|. 1848609 | 2.85,'!. ‘arg79a 
54} 848780 | 2.85 380373 
55 | 848951 | 2.85 380955 
56 | 849122 | 2.85 381537 

2.85 

283 

2 85 

2 85 


| 


10. 383870 








SEER RRP PP PE Pe ooh co co 
DMWIaI-I CLOW CI GV OI WWW CO See SSSaas speseseusns| 3 


OVOTOTOV ON OVO’ OU OT OF OF OF OF OF OF 








AND EXTERNAL SECANTS jg 


























73° 
, Vers. D, 1". |) Ex. secs *D. 1% 
0 | 9.849805 | 2.85 | 10.388870 9.48 
1 .849976 | 2.85 . 884454 9.73 
2 .850147 | 2.88 . 885038 9.75 
3 .850317 || 2.85 .885623 9.7 
4 .850488 | 2.83 .886209 9.75 
5 . 850658 | 2.85 . 386794 9.77 
6 .850829 | 2.83 887380 9.78 
7 .850999 | 2.838 .387967 9.7 
8 .851169 | 2.85 . 888554 9.7 
9 .851340 | 2.88 .889141 9.7% 
10 .651510 | 2.83 6889728 9.80 
11 | 9.851680 | 2.88 | 10.390316 9.82 
12 .851850 | 2.838 .3890905 9.80 
13 .852020 | 2.83 .8914938 9.82 
14 .852190 | 2.838 .892082 9.83 
15 . 852360 | 2.83 .892672 9.83 
16 .8525380 | 2.83 .893262 9.83 
aly .852700 | 2.83 .393852 9.85 
18 .852870 | 2.83 .894443 | 9.85 
19 .858040 | 2.82 .38950384 9.85. 
20 .858209 | 2.838 .395625 9.87 


21 | 9.853379 | 2.83 | 10.896217 | 9.87 
22 | .853549 | 2.82 -896809 | 9.88 
23 | .853718 | 2.83 3897402 | 9.88 
24 | .853888 | 2.82 .097995 | 9.90 
25 | .854057 | 2.83 .898589 | 9.88 
26 | .854227 | 2.82 .899182 | 9.92 
27 | .854396 | 2.82 899777 | 9.90 
28 | .854565 | 2.83 .400371 | 9.92 











29 | .854735 | 2.82 .400966 | 9.93 
30 | .854904 | 2.82 .401562 | 9.93 

i | 9.855073 | 2.82 | 10.402158 | 9.93 
82 | .855242 | 2.82 402754 | 9.95 
83 | .855411 | 2.82 .403351 | 9.95 
84 | .855580 | 2.82 .403948 | 9.95 
35 | .855749 | 2.82 .404545 | 9.97 
36 | .855918 | 2.82 .405143 | 9.98 
87.| .856087 | 2.80 .405742 | 9.97 
88 | .856255 | 2.82 .406340 | 9.98 
39 | 1856424 | 2.82 .406939 | 10.00 
40 | .856593 | 2.82 .407539 | 10.00 
41 | 9.856762 | 2.80 | 10.408139 | 10.00 
42 | .856930 | 2.82 .408739 | 10.02 
43 | .857099 | 2.80 | ° .409340 | 10.02 
44 | .857267 | 2.82 .409941 | 10.03 
45 | 857486 | 2.80 .410543 | 10.03 
46 | .857604 | 2.80 .411145 | 10.08 
47 | .8577'72 | 2.82 .411747 | 10.05 
48 | .857941 | 2.80 412350 | 10.07 
49 | .858109 | 2.80 412954 | 10.05 
50 | .858277 | 2.80 .413557 | 10.07 
51 | 9.858445 | 2.80 | 10.414161 | 10.08 
52 858613 | 2.80 414766 | 10.08 
53 858781 | 2.80 415371 | 10.08 
5A 858949 | 2.80 415976 | 10.10 
55 859117 | 2.80 416582 | 10.12 
56 859285 | 2.80 .417189 | 10.1C 
57 859453 | 2.80 417795 | 10.12 
58 859621 | 2.78 418402 | 10.13 
59 | .859788 | 2.80 .419010 | 10.13 
68 | 9.859956 | 2.80 | 10.419618 | 10.13 


LF TABLE XXVI.—LOGARITHMIC VERSED SINES 


a 





74° 





, Vers. |D.1".| Ex. sec. | D. 1”. 


9.859956 | 2.80 | 10.419618 | 10.13 

.860124 | 2.78 420226 | 10.15 
; .860291 | 2.80 -420835 | 10.17 
860459 | 2.78 .421445 | 10.15 
860626 -422054 | 10.17 
860794 -422664 | 10.18 
.860961 423275 | 10.18 
.861128 -423886 | 10.20 
861296 .424498 | 10.20 

9 | .861463 -425110 | 10.20 
10 | .861630 425722 | 10.22 


LUO 86197 10.426335 | 10.22 
12 | .861964 -426948 | 10.23 




















56 | .869265 
57 | .869430 
58 | .869595 
59 869760 


-454888 | 10.57 
-455022 | 10.58 








-456292 | 10.60 








2.80 

2.78 

2.78 

2.80 

2.78 

2.78 

2.78 

2.78 

2.78 
13) .862131 | 2.78 427562 | 10.23 
14 862298 | 2.78 .428176 | 10.28 
15 | .862465 | 2.78 .428790 | 10.27 
16 . 862632 | 2.78 .429406 | 10.25 
17 .862799 | 2.77 .430021 | 10.27 
18 | .862965 | 2.78 .430637 | 10.27 
19 | .863132 | 2.78 .431253 | 10.28 
20 | .868299 | 2.77. .431870 | 10.30 
21 | 9.863465 | 2.78 | 10.432488 | 10.28 
22 | .863632 | 2.78 .4383105 | 10.32 
23 863799 | 2.77 483724 | 10.380 
24 | .863965 | 2.77 .434842 | 10.32 
25 .864131 | 2.78 .434961 | 10.33 
26 864298 | 2.77 .485581 | 10.33 

7 864464 | 2.77 -486201 | 10.33 

28 | .864630 | 2.78 .436821 | 10.35 
29 | .864797 | 2.77 .437442 | 10.37 
30 | .864963 | 2.77 -438064 | 10.37 
31 | 9.865129 | 2.77 | 10.488686 | 10.37 
32 .865295 | 2.77 -439308 | 10.38 
33 | .865461 | 2.77 -4389931 | 10.38 
34 865627 | 2.77 -440554 | 10.40 
35) | .865793 | 2.77 .441178 | 10.40 
36 .865959 | 2.75 -441802 | 10.42 
37 | .866124 | 2.77 442427 | 10.42 
38 | .866290 | 2.77 .443052 | 10.48 
39 | .866456 | 2.77 443678 | 10.43 
40 | .866622 | 2.75 444304 | 10.45 
41 | 9.866787 | 2.77 | 10.444981 | 10.45 
42 .866953 | 2.75 -445558 | 10.45 
43 | .867118 | 2.77 -446185 | 10.47 
44 | .867284 | 2.75 -446813 | 10.48 
45 .867449 | 2.75 -447442 | 10.48 
46 867614 | 2.77 -448071 | 10.48 
47 | .867780 | 2.75 -448700 | 10.50 
48 867945 | 2.75 -449330 | 10.52 
49 | .868110 | 2.75 -449961 | 10.52 
50 | .868275 | 2.77 -450592 | 10.52 
51 | 9.868441 | 2.75 | 10.451228 | 10.53 
52 | .868606 | 2.75 -451855 | 10.53 
53.| .868771 | 2.75 -452487 | 10.55 
54 868986 | 2.73 -453120 | 10.57 
55 | .869100 | 2.75 .453754 | 10.57 

a 

2 

2 

2 

2 


75 
5 
U5 -455657 | 10.58 
13 
5 


60 | 9.869924 10.456928 | 10.60 





























75° 

Vers. |D.1".| Ex. sec. [D. 1° 
0 | 9.869924 | 2.75 |10.456928 | 10.60 
1 .870089 | 2.73 .457564 | 10.62 
2 .870253 | 2.75 .458201 | 10.63 
3 .870418 | 2.73 .458839 | 10.62 
4 . 870582 | 2.75 459476 | 10.65 
5 .870747 | 2.73 .460115 | 10.65 
6 .870911 | 2.75 .460754 | 10.65 
v4 .871076 | 2.73 .461393 | 10.67 
8 .871240 | 2.73 .462033 | 10.67 
9 .871404 | 2.73 .462673 | 10.68 
10 .871568 | 2.73 .463814 | 10 70 
11 | 9.871732 | 2.73 |10.463956 | 10.70 

{2 .871896 | 2.73 .464598 | 10.7 
13 .872060 | 2.73 .465240 | 10.72 
14 - 872224 | 2.73 465883 | 10.73 
15 . 872388 | 2.73 .466527 | 10.73 
16 .872552 | 2.7 .467171 | 10.73 

1% .872716 | 2.7 .467815 | 10.7 
18 .872880 | 2.72 .468460 | 10.77 

19 .873043 | 2.73 .469106 | 10.7 
20 .873207 | 2.73 .469752 | 10.77 

21 | 9.873871 | 2.72 |110.470398 | 10.7 
22 .873584 | 2.7 .471045 | 10.80 
93 . 878698 | 2.7% .471693 | 10.80 
24 .873861 | 2.73 .472341 | 10.82 
be .874025 | 2.7% .472990 | 10.82 
26 . 874188 | 2.7% .473639 | 10.83 
27 .874851 | 2.7: .474289 | 10.83 
28 874515 | 2.72 .474939 | 10.85 
29 .874678 | 2.7 .475590 | 10.87 
380 .874841 | 2.72 .476242 | 10.85 
31 | 9.875004 | 2.72 |10.476893 | 10.88 
82 S7o167| 27s .477546 | 10.88 
33 .875330 | 2.72 .478199 | 10.88 
34 .875493 | 2.7 -478852 | 10.90 
' 85 .875656 | 2.72 .479506 | 10.92 
36 .875819 | 2.72 .480161 | 10.92 
ot .875982 | 2.72 .480816 | 10.93 
38 876145 | 2.7% -481472 | 10.93 
39 .876308 | 2.7 .482128 | 10.95 
40 .876470 | 2.7 482785 | 10.95 
41 | 9.876633 | 2.72 |10.483442 | 10.97 
42 .876796 | 2.7 .484100 | 10.98 
43 .876958 | 2.72 .484759 | 10.98 
44 877121 | 2.70 485418 | 10.98 
45 Mevifice) PM .486077 | 10.98 
46 877445 | 2.7% .486738 | 11.00 
7 .877608 | 2.7 .487398 | 11 02 
48 TTT eT. .488059 | 11.03 
49 Meavereysspes | 24 ire .488721 | 11.05 
50 .878095 | 2.70 .489384 | 11.05 
51 | 9.878257 | 2.70 |10.490047 | 11.05 
52 .878419 | 2.7 .490710 | 11.07 
be .878581 | 2.70, .491874 | 11.08 
54 .8787438 | 2.7 -492039 | 11.08 
515) .878905 | 2.70 .492704 | 11.10 
56 .879067 | 2.70 -493870 | 11.10 
57 - 879229 | 2.68 .494036 | 11.12 
58 .879390 | 2.70 .494708 | 11.18 
59 .879552 | 2.70 .495371 | 11.13 
60 | 9.879714 | 2.70 110.496039 | 11.13 





ay 
SOMIMTIRWWHO | = 








Vers. 





9.879714 


9.881490 
881651 
“881812 
"881973 
882134 
882295 
882456 
882617 
882777 
(882938 


9.883099 
883260 


883420 - 


.883581 
883741 
883902 
884062 
884223 
884383 
884543 


9.884703 
884864 
885024 
885184 
885344 
885504 
885664 
. 885824 
885983 
886143 


9.886303 
. 886463 
886622 
886782 
886941 
887101 
. 887260 
. 887420 
887579 
887739 

9.887898 
888057 
888216 
888375 

888534 
888693 
888852 
. 889011 
889170 
9.889329 








for) 
ie 2) 





DADA AAADA2DAIAANWA 
NIINDWNIBDNDON 


IND 





WW WWW WWNWWD WNWWWNWNWWWWW f 





WWW WNWWWWWW 
RH ARIANA RTAR ARQ RRTAVWVWNR 


DDD. HAADAAAAAARDD DRUM DWAIS 


AND EXTERNAL SECANTS 


76° 


Ex. sec. 


10.496039 
-496707 
497377 
.498047 
.498717 
.499388 
.500060 
900732 
501405 
502078 
502752 


10.503426 
504102 
504777 
505454 
.506131 
506808 
597486 
508165 
508844 
909524 


10.510205 
510886 
511568 

1512250 
(512933 
‘513617 
‘514301 
514986 
‘515672 
(516358 


10.517045 
017732 
918420 
519109 
.919798 
520488 
521179 
.521870 
922562 
523254 

10.528947 


530215 
10.530914 
531614 
532315 
> 533017 
533719 
534422 
535126 
535830 
536535 








forpForkorgorgor 
Gi Sr St Gt St Or oe 


10.537241 








CANOUIP WMHS | 




















173 





Vers. 


9.889329 
"889488 
889647 
“889805 
'889964 
890123 
‘890281 
"890440 
:890598 
‘890757 
"890915 


9.891073 
891282 
.891390 
891548 
891706 
.891864 
892022 
892180 
892338 
892496 


9 892654 
892812 
.892969 
893127 
893285 
893442 
893600 
893758 
.893915 
894072 


9.894230 
894387 
894544 
.894702 
894859 
895016 
895173 
895330 
895487 
895644 


9.895801 
895958 
896115 
896272 
896428 
896585 
896742 
896898 
897055 
897211 


9.897368 
897524 
897680 
897837 
897993 
898149 
898305 
898461 
898618 





ct i 


ete 





Wwwwwwwwwww 
PO AQR2®BRRADS 220000890 
Gd G2 G9 G9 G9 G9 C9 OD OD GL GI GD Od HO HO Gd OUT 


=> 
CO 0 OO 


WWWWNWWWWWW WWWWWIOWNWWYW 
SS? So Ss S32 G2 Sd 


WWWWW WHWNWWWWW 


WNW WW WNWWWWW 
DP>DVPDPVWHYD ADAIAAIWWDAIS D 


SS 


WOWNWNW WOWWWN 


Ooo Wow 


VWI WW WWWwwOWW 
ODS OSs 





SS 
o 


2.60 


| 


Exasecr | Ds l*, 


10.587241 | 11.77 
.O387947 | 11.78 
538654 | 11.80 
.539362 | 11.82 
.540071 | 11.82 
.540780 | 11.83 
-541490 | 11.83 
542200 | 11.85 
042911 | 11.87 
543623 | 11.88 
.544836 | 11.88 


10.545049 | 11.90 
.545763 | 11.90 
546477 | 11.98 
.547193 | 11.98 
.547909 | 11.95 
548626 | 11.95 
.549343 | 11.97 
.550061 | 11.98 
.550780 | 12.00 
.551500 | 12.00 


10552220 | 12.02 
.002941 | 12.03 
.553663 | 12.03 
.554885 | 12.07 
.555109 | 12.07 
.555833 | 12.07 
.556557 | 12.10 
.957283 | 12.10 
.558009 | 12.12 
.558736 | 12.12 

10.559463 | 12.15 
.560192 | 12.15 
.560921 | 12.17 
.561651 | 12.17 
562881 | 12.20 
.563113 | 12.20 
.563845 | 12.20 
-564577 | 12.23 
-565311 | 12.23 
-566045 | 12.27 


10.566781 | 12.25 
.567516 | 12.28 
.568253 | 12.28 
.568990 | 12.32 
569729 | 12.32 
.570468 | 12.382 
971207 | 12.35 
.571948 | 12.35 
.572689 | 12.37 
573431 | 12.38 


10.574174 | 12.38 

















580145 | 12.50 


9.898774 | 2.60 |10.580805 | 12.50 


174 TABLE XXVI.—LOGARITHMIC VERSED SINES 























.626650 | 13.37 
10.627452 | 13.40 


59 | .907898 
60 | 9.908051 


re 


78° 
|) Vers.» | D, 1!) Hix, see; (| D. 1". 
0 | 9.898774 | 2.60 | 10.580895 | 12.50 
1 .898930 | 2.60 -581645 | 12.53 
2 899086 | 2.58 582397 | 12.58 
3 899241 | 2.60 .083149 | 12.57 
4 899397 | 2:60 .583903 | 12.57 
5 899553 | 2.60 .084657 | 12.57 
6 899709 | 2.60 .085411 | 12.60 
7 899865 | 2.58 .586167 | 12.60 
8 .900020 | 2.60 .586923 | 12.63 
9 .900176 | 2.58 .587681 | 12.63 
10 .900831 | 2.60 .588489 | 12.65 
11 | 9.900487 | 2.58 | 10.589198 | 12.65 
12 . 900642 | 2.60 .589957 | 12.68 
13 .900798 | 2.58 -590718 | 12.68 
14 . 900958 | 2.58 .591479 | 12.72 
15 .901108 | 2.60 092242 | 12.72 
16 .901264 | 2.58 .598005 | 12.73 
17 .901419 | 2.58 -5938769 | 12.73 
18 .901574 | 2.58 .594533 | 12.77 
19 | .901%729 | 2.58 .595299 | 12.78 
20 .901884 | 2.60 .596066 | 12.78 
21 | 9.902040 | 2.58 | 10.596833 .| 12.80 
22 .902195 } 2.58 .597601 | 12.82 
25 .902850 | 2.57 * 59887 12.83 
24 | .902504 | 2.58 599140 | 12.85 
25 .902659 | 2.58 599911 | 12.85 
26 .902814 | 2.58 600682 | 12.88 
27 . 902969 | 2.58 601455 | 12.88 
28 .9038124 | 2.57 602228 | 12.92 
29 .908278 | 2.58 -603003 | 12.92 
80 .908433 | 2.58 .608778 | 12.93 
31 | 9.908588 | 2.57 | 10.604554 | 12.95 
82 .903742 | 2.58 .605831 | 12.95 
33 .908897 | 2.57 -606108 | 12.98 
34 -904051 | 2.58 .606887 | 13.00 
35 .904206 | 2.57 .607667 | 13.00 | 
36 .904360 | 2.57 .608447 | 13.02 
37 .904514 | 2.57 -609228 | 138.03 
38 -904668 | 2.58 .610010 | 13.07 
39 .904823 | 2.57 .610794 | 138 07 
40 | .904977 | 2.57 .611578 | 13.08 
41 | 9.905131 | 2.57 | 10.612368 | 13.08 
42 .905285 | 2.57 .613148 | 13.12 
48 .905439 | 2.57 .613935 | 13.13 
44 .905593 | 2.57 .614723 | 13.13 
45 905747 | 2.57 .615511 | 13.17 
46 .905901 | 2.57 -616301 | 18.17 
47 .906055 | 2.57 .617091 | 13.20 
48 .906209 | 2.57 -617883. | 18.20 
49 .906368 | 2.55 -618675 | 13.22 
50 .906516 | 2.57 .619468 | 18.23 
51 | 9.906670 | 2.57 | 10.620262 | 13.25 
52 .906824 | 2.55 -621057 | 13.27 
53 .906977 | 2.57 .621853 | 18.28 
54 .9071381 | 2.55 .622650 | 13.30 
55 .907284 | 2.57 -6238448 | 13.382 
56 .907488 | 2.55 .624247 | 13.38 
57 .907591 | 2.55 .625047 | 18.35 
58 907744 | 2.57 -625848 | 13.37 
2.65 
2.55 








eee Rt 
OD Bomrsmawwns| > 


79° 





Vers. |D.1’.| Ex. sec. |D. 1’, 

















9.908051 | 2.55 |10.627452 | 13.40 
-908204 | 2.55 | .628256 | 13.40 
-908357 | 2.57 | .629060 | 13.43 
.908511 | 2.55 | .629866 | 13.45 
.908664 | 2.55 | .630673 | 13.45 
-908817 | 2.55 | .681480 | 13.48 
908970 | 2.55 | .632289 | 13.48 
.909123 | 2.55 | .633098 | 13.52 
909276 | 2.53 | .633909 | 13.52 
.909428 | 2.55 | .634720 | 138.55 
.909581 | 2.55 | .685533 | 18.55 


9.909784 | 2.55 |10.636346 | 13.58 
.909887 | 2.53 | .637161 | 138.58 
-910039 | 2.55 | .637976 | 13.60 
.910192 | 2.55 | .638792 | 138.63 
-9103845 | 2.53 | .639610 | 13.63 
-910497 | 2.55 | .640428 | 13.67 
-910650 | 2.53 | .641248 | 13.67 
-910802 | 2.55 | .642068 | 13.70 
-910955 | 2.53 | .642890 | 13.72 
-911107 | 2.53 | .648713 | 13.72 

9.911259 | 2.55 |10.644536 | 13.75 
.911412 | 2.53] .645361 | 138.75 
.911564 | 2.53 | .646186 | 13.78 
-911716 | 2.53 | .647013 | 18.80 ~ 
-911868 | 2.53 | .647841 | 13.82 
.912020 | 2.53 | .648670 | 13.82 
912172 | 2.53 | .649499 | 13.85 
912824 | 2.53 | .650830 | 13.87 


912476 | 2.53 | .651162 | 13.88 
912628 | 2.53 | .651995 | 13.90 


9.912780 | 2.53 |10.652829 | 13.92 
.912982 | 2.53 | .653664 | 13.95 
918084 | 2.52 | .654501 | 13.95 
-913285 | 2.53 | .655338 | 13.97 
-913387 | 2.53] .656176 | 14.00 
-918539 | 2.52 | .657016 | 14.00 
.913690 | 2.53 | .657856 | 14.03 
-918842 | 2.52 | .658698 | 14.03 
.918993 | 2.53 | .659540 | 14.07 
914145 | 2.52 | .660384 | 14.08 


9.914296 | 2.53 |10.661229 | 14.10 
.914448 | 2.52 | .662075 | 14.12 
914599 | 2.52 | .662922 | 14.13 
914750 | 2.53 | .663770 | 14.15 
-914902 | 2.52 | .664619 | 14.18 
-915053 | 2.52 | .665470 | 14.18 . 
915204 | 2.52 | .666321 | 14.22 





915855 | 2.52 | .667174 | 14.23 
-915506 | 2.52 | .668028 | 14.25 
-915657 | 2.52 | .668883 | 14.27 

9.915808 | 2.52 |10.669739 | 14.28 
.915959 | 2.52 | .670596 | 14.30 
.916110 | 2.52 | .671454 | 14.38 
916261 | 2.52 | .672314 | 14.33 
.916412 | 2.50 | .678174 | 14.37 
.916562 | 2.52 | .674036 | 14.38 
916713 | 2.52 | .674899 | 14.40 
.916864 | 2.50 | .675763 | 14.42 
.917014 | 2.52 | .676628 | 14.45 

9.917165 | 2.52 


10.677495 | 14.45 . 





AND EXTERNAL SECANTS 175 
















































































80° 81° 
4 Vers, ~} Dal’.| Ex. 86c. | D. 1’, e Vers. |D.1".| Ex. sec. | D. 1". 
O | 9.917165 | 2.52 | 10.677495 | 14.45 0 | 9.926119 | 2.47 |10.731786 | 15.78 
1 .9173816 | 2.50 .678362 | 14.48 1 926267 | 2.47 .782738 | 15.78 
2 .917466 | 2.50 679281 | 14.50 2 .926415 | 2.45 . 738680 | 15.83 
3 917616 4 2.52 680101 | 14.52 3 . 926562 | 2.47 . 784630 | 15.83 
4 .917767 | 2.50 -680972 | 14.55 4 .926710 | 2.47 .7385580 | 15.87 
5 MOTT 1-2:.52 681845 | 14.55 5 | .926858 | 2.47 . 736532 | 15.90 
6 .918068 | 2.50 .682718 | 14.58 6 .927006 | 2.45 . (37486 | 15.92 
Hi .918218 | 2.50 .§88593 | 14.60 Vi .927153 | 2.47 .788441 | 15.95 
8 .918868 | 2.50 684469 | 14.62 8 .9273801 | 2.45 . 7393898 | 15.97 
9 .918518 | 2.50 .685346 | 14.63 ||. 9 927448 | 2.47 . 740856 | 16.00 
10 .918668 | 2.50 .686224 | 14.67 || 10 .927596 | 2.45 .741316 | 16.02 
‘11 | 9.918818 ; 2.50 | 10.687104 | 14.68 || 11 | 9.927748 | 2.47 110.742277 | 16.08 
12 .918968 | 2.50 .687985-| 14.7 12 .927891 | 2.45 . 748239 | 16.08 
13 .919118 | 2.50 -688867 | 14.72 || 13 928088 | 2.45 744204 | 16.08 
14 .919268 | 2.50 -689750 | 14.73 || 14 .928185 | 2.47 .745169 | 16.13 
15 -919418 | 2.50 -690634 | 14.77 || 15 . 928838 | 2.45 .746137 | 16.13 
16 .919568 | 2.50 .691520 | 14.78 || 16 . 928480 | 2.45 -747105 | 16.18 
17 .919718 |. 2.50 -692407 | 14.80 || 17 928627 | 2.45 .748076 | 16.20 
18 .919868°| 2.50 .693295 | 14.83 || 18 928774 | 2.45 . 749048 | 16.22 
19 .920018 | 2.48 694185 | 14.83 || 19 928921 | 2.45 | .750021 | 16.25 
20 . 920167 | 2.50 .695075 | 14.87 || 20 .929068 | 2.45 .750996 | 16.28 
21 | 9.920817 | 2.48 | 10.695967 | 14.90 || 21 | 9.929215 | 2.45 |10.751978 | 16.30 
22 .920466 | 2.50- .696861 | 14.90 || 22 . 929862 | 2.45 . 752951 -| 16.388 
23 .920616 | 2.50 -697755 | 14.93 || 23 .929509 | 2.45 .758981 | 16.35 
24 | .920766 | 2.48 | .698651 | 14.95 || 24 .929656 | 2.45 .754912 | 16.38 
Rd -920915 | 2.48 .699548 | 14.97 || 25 .929808 | 2.45 .755895 | 16.42 
26 .921064 | 2.50 . 700446 | 15.00 26 .929950 | 2.45 . 756880 | 16.43 
27 .921214 | 2.48 701346 | 15.02 || 27 .980097 | 2.43 .757866 | 16.47 
28 .921363 | 2.48 702247 | 15.03 || 28 .980243 | 2.45 . 758854 | 16.50 
29 .921512 | 2.50 . 703149 | 15.05 |} 29 .9380890 | 2.45 . 759844 | 16.52 
30 .921662 | 2.48 704052 | 15.08 || 380 . 9805387 | 2.43 . 60885 | 16.58 
81 | 9.921811 | 2.48 | 10.704957 | 15.10 || 31 | 9.930688 | 2.45 |10.761827 | 16.58 
32 . 921960 | 2.48 . 705863 | 15.18 || 82 .930880 | 2.43 . 762822 | 16.60 
33 . 922109 | 2.48 706771 | 15.15 || 83 .980976 | 2.45 .763818 | 16.62 
34 . 922258 | 2.48 - 707680 | 15.17 || 84 .981128 | 2.48 .764815 "| 16.67 
35 . 922407 | 2.48 - 708590 | 15.18 || 35 .931269 | 2.45 . 765815 | 16.68 
36 . 922556 | 2.48 - 709501 | 15.22 || 36 .931416 | 2.43 .766816 | 16.72 
37 . 922705 | 2.48 (10414 | 15:23 |1.87 .981562 | 2.48 767819 | 16.73 
38 . 922854 | 2.48 «711828 | 15.25 || 38 .931708 | 2.45 . 768823 | 16.77 
39 .928003 | 2.48 712248 | 15.28 || 39 .931855 | 2.43 . 769829 | 16.80 
40 923152 | 2.48 .718160 | 15.80 || 40 .982001 | 2.43 . 770887 | 16.82 
41 | 9.923301 | 2.47 | 10.714078 | 15.33 || 44 | 9.932147 | 2.43 |10.'771846 | 16.87 | 
42 .923449 | 2.48 ~714998 | 15.85 || 42 . 982293 | 2.438 772858 | 16.87 | 
43 . 923598 | 2.48 715919 | 15.37 || 438 .982489 | 2.48 . 7738870 | 16.92 
44 923747 | 2.47 .716841 | 15 88 || 44 .982585 |°2.48 .774885 | 16.95 
45 . 923895 | 2.48 717764 | 15.42 || 45 .9827381 | 2.438 775902 | 16.97 
46 .924044 | 2.47 ~718689 | 15.45 || 46 982877 | 2.43 .776920 | 17.00 
Av .924192 | 2.48 -719616 | 15.45 || 47 .988023 | 2.43 777940 | 17.02 
48 .9243841 | 2.47 .720548 | 15.48 || 48 .933169 | 2.48 .778961 | 17.07 
49 -924489 | 2.47 «(21472 | 15.52 |) 49 .9388315 | 2.42 -779985 | 17.08 
50 . 9246387 | 2.48 (22408 | 15.53 |; 50 | .9838460 | 2.438 .781010 | 17.12 
51 | 9.924786 | 2.47 | 10.7238385 | 15.55 || 51 | 9.933606 | 2.43 |10.782087 | 17.13 
52 .924984 | 2.47 - 724268 | 15.58 || 52 .9838752 | 2.42 .783065 | 17.18 
53 .925082 | 2.48 .7252038 | 15.60 || 53 .933897 | 2.48.) .784096 | 17.20 
54 .925231 | 2.47 fs .726139 | 15.63 || 54 .984043 | 2.43 785128 | 17.28 
55 | .925379 | 2.47 727077 | 15.65 || 55 .934189 | 2.42 .786162 | 17.27 
56 .925527 | 2.47 - 728016 | 15.67 || 56 . 934334 | 2 43 .787198 | 17.30 
BY .925675 | 2.47 .728956 | 15.70 || 57 .934480 | 2.42 .788236 | 17.38 
58 .925823 | 2.47 .729898 | 15.73 || 58 . 984625 | 2.42 789276 | 17.385 
59 925971 | 2.47 . 730842 | 15.7% 59 .934770 | 2.48 .7903817 | 17.40 
60 | 9.926119 | 2.47 | 10.731786 | 15.78 || 60 | 9.934916 | 2.42 110.791361 | 17.42 


a 


176 TABLE XXVI.—LOGARITHMIC VERSED SINES 



























































82° 83° 
, Vers. GiiD. dee x Secr Ds 125677 Vers. |D.1".| Ex. sec. | D. 1’. 
O | 9.934916 | 2.42 | 10.791361 | 17.42 0 | 9.948559 | 2.38 |10.857665 | 19.55 
1 .935061 | 2.42 - 792406 | 17.45 i .948702 | 2.38 -858888 | 19.58 
2 . 9385206 | 2.43 .7934538 | 17.48 2 - 948845 | 2.37 .860013 | 19.63 
3 .985352 | 2.42 ~ 794502 | 17.50 3 .9438987 | 2.38 -861191 | 19.67 
4 .9385497 | 2.42 ~ 795552 | 17.55 4 .944130 | 2.38 .8623871 | 19.72 
5) . 9385642 | 2.42 .796605 | 17.58 5 -944273 | 2.37 .863554 | 19.75 
6 .9385787 | 2.42 .797660 | 17.60 6 .944415 | 2.38 .864739 | 19.80 
7 . 985982 | 2.42 -798716 | 17.63 a .944558 | 2.37 .865927 | 19.83 
8 .9386077 | 2.42 .799774 | 17.68 8 - 944700 | 2.38 .867117 | 19.88 
|9 . 9386222 | 2.42 .800885 | 17.70 9 - 944843 | 2.37 .868310 | 19.92 
10 | .9386867 | 2.42 .801897 | 17.73 || 10 | .944985 | 2.37 | .869505 | 19.97 
11 | 9.986512 | 2.42 | 10.802961 | 17.77 || 11 | 9.945127 | 2.88 |10.870703 | 20.00 
12 .936657 | 2.40 -804027 | 17.80 || 12 -945270 | 2.37 .871903 | 20.05 
13 .9386801 | 2.42 -805095 | 17.83 || 13 .945412 | 2.37 .873106 | 20.10 
14 .936946 | 2.42 .806165 | 17.87 || 14 .945554 | 2.37 .874812 | 20.13 
15 .987091 | 2.42 .807237 | 17.90 || 15 . 945696 | 2.37 .875520 | 20.18 
16 . 9372386 | 2.40 .808311 | 17.93 || 16 - 945838 | 2.38 .8767381 | 20.23 
17 .937380 | 2.42 .809887 | 17.97 || 17 -945981 | 2.37 .877945 | 20.27 
18 .937525 | 2.40 .810465 | 18.00 || 18 - 946123 | 2.37 .879161 | 20.80 
19 .937669 | 2.42 .811545 | 18.03 19 .946265 | 2.37 .880379 | 20.37 
20) .9387814 | 2.40 .812627 | 18.07 || 20 .946407 | 2.37 | .881601 | 20.40 
21 | 9.937958 | 2.42 | 10.818711 | 18.10 || 21 | 9.946549 | 2.35 |10.882825 | 20.45 
22 .938108 | 2.40 .814797 | 18.13 || 22 .946690 | 2.37 884052 | 20.48 
23 . 988247 | 2.40 .815885 | 18.17 || 23 - 946882 | 2.37 885281 | 20.55 
24 .988391 | 2.42 .816975 | 18.20 || 24 -946974 | 2.387 886514 | 20.58 
25 . 988536 | 2.40 .818067 | 18.23 || 25 .947116 | 2.37 .887749 | 20.62 
26 | .938680 | 2.40 .819161 | 18.27 || 26 | .947258 | 2.35 .888986 | 20.68 
if . 9388824 | 2.40 so20200 VW lSx82 eo? .947399 | 2.37 - 890227 | 20.72 
28 . 938968 | 2.40 .821356 | 18.338 || 28 Oa AL | 223 .891470 | 20.77 
29 .939112 | 2.38 .822456 | 18.38 || 29 .947683 | 2.35 .892716 | 20.82 
80 | .989257 | 2.40 .823559 | 18.42 || 80 | .947824 | 2.37 .893965 | 20.87 
81 | 9.939401 | 2.40 | 10.824664 | 18.43 || 31 | 9.947966 | 2.35 |110.895217 | 20.92 
2 ~939D45 || 2.38 .825770 | 18.48 || 32 .948107 | 2.37 .896472 | 20.95 
33 | .939688 | 2.40 .826879 | 18.52 || 33 .948249 | 2.35 .897729 | 21.00 
34 . 9398382 | 2.40 -827990 | 18.57 || 34 - 948390 | 2.35 .898989 | 21.07 
3D . 9389976 | 2.40 .829104 | 18.58 || 35 .948531 | 2.37 . 900253 | 21.10 
36 .940120 | 2.40 .8380219 | 18.63’ || 36 .948673 | 2.35 -901519 | 21.15 
37 . 940264 | 2.40 .83813387 | 18.65 |! 37 .948814 | 2.35 .902788 | 21.20 
388 .940408 | 2.38 .882456 | 18.70 || 38 .948955 | 2.35 .904060 | 21.25 
39 .940551 | 2.40 , 833578 | 18.75 || 89 - 949096 | 2.35 . 905385 | 21.30 
40 .940695 | 2.40 .8347038 | 18.77 || 40 949287 | 2.37 .906613 | 21.33 
41 | 9.940839 | 2.88 | 10.835829 | 18.80 || 41 | 9.949379 | 2.35 |10.907893 | 21.40 
42 - 940982 | 2.40 .886957 | 18.85 || 42 . 949520 | 2.35 .909177 | 21.45 
43 - 941126 | 2.38 .838088 | 18.88 || 43 -949661 | 2.35 .910464 | 21.50 
44 . 941269 | 2.40 .839221 | 18.93 || 44 .949802 | 2.35 .911754 | 21.55 
45 .941413 | 2.38 .840857 | 18.95 || 45 . 949948 | 2.33 .913047 | 21.60 
46 . 941556 | 2.38 .841494 | 19.00 || 46 .950088 | 2.35 .914348 | 21.65 
ve .941699 | 2.40 .842634 | 19.038 || 47 5950224 | 2.35 .915642 | 21.7 
48 - 941843 | 2.38 .8438776 | 19.08 || 48 .950865 | 2.85 .916944 | 21.75 
49 .941986 | 2.38 .844921 | 19.12 || 49 .950506 | 2.35 -918249 | 21.82 
50 . 942129 | 2.38 .846068 | 19.15 || 50 . 950647 | 2.338 -919558 | 21.85 
51 | 9.942272 | 2.38 | 10.847217 | 19.18 || 51 | 9.950787 | 2.35 |10.920869 | 21.92 
52 .942415 | 2.40 .848368 | 19.23 || 52 .950928 | 2.35 .922184 | 21.97 
53 .942559 | 2.38 . 849522 | 19.27 || 53 .951069 | 2.33 . 928502 | 22.02 
54 . 942702 | 2.38 .850678 | 19.380 || 54 -951209 | 2.85 - 924823 | 22.07 
55 .942845 | 2.38 .851886 | 19.35 || 55 .951350 | 2.33 .926147 | 22.13 
56 - 942988 | 2.38 .852997 | 19.40 || 56 .951490 | 2.35 927475 | 22.17 
57 . 943131 | 2.37 .854161 | 19.42 || 57 .951631 | 2.33 -928805 | 22.23 
58 . 9438273 | 2.38 .8553826 | 19.47 || 58 951771 | 2.33 .930139 | 22.30 
59 . 948416 | 2.38 .856494 | 19.52 || 59 .951911 | 2.35 .9381477 | 22.33 
60 ! 9.948559 | 2.38 | 10.857665 | 19.55 |! 60 | 9.952052 | 2.33 110.9382817 | 22.40. 








a 


AND EXTERNAL SECANTS br Srg 





84° 


9.952052 | 2.33 
.952192 | 2.33 
.952332 | 2.35 
952473 | 2.33 
.952613 | 2.33 
-952753 | 2.33 
952893 | 2.33 
953033 | 2.33 
9538173 | 2.33 
953318 | 2.33 
.958453 | 2.33 


9.953593 | 2.32 
953732 | 2.33 
958872 | 2.33 
.954012 | 2.33 
.954152 | 2.382 
954291 | 2.33 
954431 | 2.33 
.954571 | 2.32 

19 .954710 | 2.33 

20 954850 | 2.32 


21 | 9.954989 | 2.33 
22] .955129 | 2.382- 
23 |. .955268 | 2.382 
24 | .955407 | 2.38 
25 | .955547 | 2.82 
26 | .955686 | 2.32 
27 | .955825 | 2.382 
28 | .955964 | 2.32 
29 .956103 | 2.388 
380 | .956243 | 2.32 


31 | 9.956382 | 2.32 
32 .956521 | 2.382 
33 -956660 | 2.32 
384 | .956799 | 2.30 


oe a 
[eer SonmIrMAwwHo | - 














35. | .956987 | 2.382 
36 -957076 | 2.32 
387 | .957215 | 2.32 
38 | .9573854 | 2.32 
39 -957493 | 2.30 
40 957631 | 2.32 
41 | 9.957770 | 2.32 
2 .957909 | 2.30 
43 | .958047 | 2.382 
44 | .958186 | 2.30 
45 . 958324 | 2.32 
46 .958463 | 2.30 
47 .958601 | 2.30 
48 .958739 | 2.32 
49 | .958878 | 2.30 
50 .959016 | 2.30 
51 | 9.959154 | 2.30 
52 | 959292 | 2.382 
53 | .959431 | 2.30 
54 | ».959569 | 2.30 
55 | .959707 | 2.380 
56 | .959845 | 2.30 
57 | .959983 | 2.30 
58 | .960121 | 2.30 
59 | .960259 | 2.30 
60 | 9.960397 | 2.30 


Vers. ». 1”.| Ex. sec. 


10.932817 
.934161 
935508 
9386859 
938213 
-939570 
- 940931 
- 942296 
943663 
945034 
. 946409 


10.947787 
.949169 
950554 
951943 
953336 
954732 
956132 
957535 
958942 
-960353 


10.961767 
963186 
964608 

_ .966034 
967463 
968897 
970334 
ALT 
973221 

74670 


10.976128 
977580 
979041 
980506 
981975 
983448 
984926 
986407 
987893 
- 989383 


10.990877 
«99237 
998877 
995384 
.996895 
.998411 
.999931 

11.001455 
002984 
004517 

11.006055 
007597 
-009144. 


*s 010695 


012251 
0188114 
015377 
016947 
018521 
11.020101 























= ~ 
SCODIRDBUIRWWHO 


Vers. 


9.960397 


960535 
. 960672 
. 960810 
960948 
961086 
961228 
961361 
961498 
. 961636 
961773 


9.961911 
962048 
. 962186 
- 962328 
- 962460 
962597 
962735 
962872 
. 963009 
963146 


9.963283 


"964515 


9.964651 
.964788 
964924 
. 965061 
965197 
965334 
. 965460 
. 965607 
965743 
965879 


9.966016 
-966152 
966288 
966424 
996560 
. 966696 
. 966832 
. 966968 
967104 
967240 

9.967376 
967512 
967647 
967783 
. 967919 
- 968054 
.968190 
- 968326 
. 968461 

9.968597 


85° 


Dislia xe seer Ds t. 


2.30 /11.020101 | 26.40 
2.28 | .021685 | 26.48 
2.380 | .028274 | 26.57 
2.30 | .024868 | 26:65 
2.380 | .026467 | 26.73 
2.28 -028071 | 26.80 
2.30 029679 | 26.90 
2.28 .031293 | 26.98 
2.30 | .082912 | 27.07 
2.28 -034536 | 27.18 
2.80 036164 | 27.28 


2.28 j11.037798 | 27.33 




















2.30 | 039438 | 27.40 
2.28 | 041082 | 27.50 
2.28] 042732 | 27.58 
2.28 | .044387 | 27.67 
2.30 | 1046047 | 27.77 
98 | .047713 | 27.85 
28 | .049384 | 27.93 
28 | .051060 | 28.03 
28 | 052742 | 28.13 
28 |11.054430 | 28.22 
28 | .056123 | 28.30 
28 | .057821 | 28.40 
28 | .059525 | 28.50 
28 | .061235 | 28.60 
27 | .062951 | 28.68 
28 | 064672 | 28.78 
28 | .066399 | 28.88 
28 | 068132 | 28.98 
27 | .069871 | 29.08 


& 


11.071616 | 29.18 
.073367 | 29.28 
075124 | 29.38 
076887 | 29.48 
.078656 | 29.58 
080431 | 29.68 
082212 | 29.80 
-084000 | 29.90 
-085794 | 80.00 
087594 | 380.12 


2.27 |11.089401 | 30.22 
2.27 .091214 | 80.32 
2.27 | .093083 | 80.48 
2.27 | .094859 | 30.55 
2.27 | .096692 | 80.67 
2.27 | .098582 | 80.77 
2.27 | .100878 | 30.87 
2.27 | .102230 | 81.00 
2.27 .104090 | 81.12 
2.27 | .105957 | 31.22 


2.27 |11.107830 | 31.35 
2.25 | .109711 | 81.45 
212018 . 111598") 31.08 
2.27} .113493 | 31.68 
2.25 | .115394 | 31.82 
2.27} .117803 | 31.98 
2.2 .119219 | 32.07 
2.25 121143 | 32.18 
2.27 | .123074 | 82.30 
2.25 |11.125012 | 382.43 


Wwwwnwnwwunwe 
D-IIDID IDR 








178 TABLE XXVI.—LOGARITHMIC VERSED SINES 





















































86° 87° 

, Vers. —|'D. .1%.|. Exesee; | D1", ||’ Vers, | D. 1" 
0 | 9.968597 | 2.25 | 11.125012 | 32.48 0 | 9.976654 | 2.23 
1 . 968732 | 2.27 . 126958 | 32.55 1 976788 | 2.22 
Q . 968868 | 2.25 .128911 | 82.70 2 976921 | 2.22 
8 . 969008 | 2.25 .1380873 | 82.80 3 .977054 | 2.22 
4 .969188 | 2,27 ~132841 | 32.95 4 977187 | 2.22 
5 .969274 | 2.25 .134818 | 33.07 5 . 9773820 | 2.20 
6 . 969409 | 2.25 . 136802 | 83.22 6 .977452 | 2.22 
q . 969544 | 2.25 .188795 | 83.33 va 977585 | 2.22 
8 .969679 | 2.25 -140795 | 83.47 8 977718 | 2.22 
9 .969814 | 2.25 . 142808 | 83.62 9 .97'7851 | 2.22 
10 . 969949 | 2.25 - 144820 | 83.73 || 10 .977984 | 2.20 
11 | 9.970084 | 2.27 | 11.146844 | 83.88 || 11 | 9.978116 | 2.22 
12 . 970220 | 2.23 .148877 | 84.02 || 12 .978249 | 2.22 
13 . 970854 | 2.25 -150918 | 84.17 || 13 . 978882 | 2.20 
14 .970489 | 2.25 - 152968 | 34.30 14 .978514 | 2.22 
15 . 970624 | 2.25 -155026 | 34.43°|) 15 .978647 | 2.20 
16 970759 | 2.25 .157092 | 34.60 || 16 978779 | 2.22 
7 . 970894 | 2.25 .159168 | 84.73 17 .978912 | 2.20 
18 .971029 | 2.25 -161252 | 84.87 18 .979044 | 2.22 
19 . 971164 | 2.28 . 163344 | 85.03 || 19 979177 | 2.20 
20 .971298 | 2.25 . 165446 | 35.17 || 20 .979309 | 2.22 
21 | 9.971438 | 2.25 | 11.167556 | 85.33 || 21 | 9.979442 | 2.20 
22 .971568 | 2.23 .169676 | 35.48 |} 22 .979574 | 2.20 
23 971702 | 2.25 -171805 | 35.63 || 23 .979706 | 2.20 
2 9718387) | 2.28 -178948 | 85.78 || 24 .979888 | 2.20 
25 971971 | 2.25 -176090 | 35.93 || 25 .979970 | 2.22 
26 .972106.| 2.23 .178246 | 86.10 || 26 .980103 | 2.20 
27 - 972240 | 2.23 .180412 | 86.27 || 27 - 980235 | 2.20 
28 . 972874 | 2.25 -182588 | 36.42 28 . 980367 | 2.20 
29 . 972509 | 2,28 -184773 | 36.58 || 29 .980499 | 2.20 
30 . 972643 | 2.28 -186968 | 36.75 || 30 . 980631 } 2.20 
31 | 9.972777 | 2.25 | 11.189173 | 36.90 || 81 | 9.980763 | 2.20 
32 . 972912 | 2,23 -191387 | 37.08 |] 82 .980895 | 2.18 
33 . 973046 | 2.23 193612 | 37.25 || 33 681026 | 2.20 
34 .973180 | 2.23 195847 | 87.42 || 34 .981158 | 2.20 
35 .978314 | 2,23 198092 | 37.58 || 35 - 981290 | 2.20 
36 .973448 | 2,23 200347 | 37.77 || 86 . 981422 | 2.20 
37 . 973582 | 2.23 202613 | 37.93 || 87 .981554 | 2.18 
38 .973716 | 2.23 204889 | 38.12 ||} 88 .981685 | 2.20 
39 .973850 | 2.23 -207176 | 38.28 || 89 .981817 | 2.20 
40 . 973984 | 2.23 .209473 | 38.47 || 40 . 981949 | 2.18 
41 | 9.974118 | 2.23 | 11.211781 | 38.67 || 41 | 9.982080 | 2.20 
42 . 974252 | 2.23 -214101 | 38.838 |} 42 - 982212 | 2.18 
43 -974386 | 2,22 -216431 | 89.03 || 43 . 9828438 | 2.20 
44 .974519 | 2.23 -218773 | 39.20 || 44 . 982475 | 2.18 
45 974653 | 2.23 -221125 | 39.42 || 45 . 982606 | 2.18 
46 974787 | 2,22 » .223490 | 39.58 || 46 . 9827387 | 2.20 
7 974920 | 2.23 -225865 | 89.80 || 47 . 982869 | 2.18 
48 . 975054 | 2.23 -228253 | 39.98 || 48 -983000 | 2.18 
49 .975188 | 2,22 .280652 | 40.18 || 49 .983131 | 2.18 
50 /9753821 | 2.28 .293063 | 40.38 || 50 . 988262 | 2.20 
51 | 9.975455 | 2.22 | 11.235486 | 40.58 || 51 | 9.988394 | 2.18 
52 .975588 | 2,23 .237921 | 40.78 || 52 983525 | 2.18 
53 -975722 | 2,22 .240368 | 41.00 || 53 983656 | 2.18 
54 975855 | 2.22 .242828 | 41.20 || 54 983787 | 2.18 
tals) .975988 | 2.23 .245300 | 41.42 || 55 983918 | 2.18 
56 . 976122 | 2,22 .247785 | 41.63 || 56 984049 | 2.18 
57 .976255 | 2,22 . 250283 | 41.83 || 57 984180 | 2.18 
58 . 976388 | 2.22 .2527938 | 42.07 || 58 984311 | 2.18 
59 .976521 | 2.22 .2553817 | 42.28 || 59 .984442 | 2.18 
60 | 9.976654 | 2.28 9.984573 | 2.17 


11.257854 | 42.52 || 60 





.| Ex. sec. 


11.257854 
-260405 
262969 
265546 
268138 
270748 
»273368 


275996 | 


278645 
281308 
. 283986 


11.286679 
289387 
.292110 
294849 
297604 
800374 
303161 
-305964 
808784 
-311620 


11.314473 
317348 
820231 
823137 
-326060 
-829001 
-331961 
-334939 
3837985 
.340951 


11.348986 
847041 
850115 
353210 
856825 
3859460 
862617 
. 3865794 
.368993 
872214 


11.375458 
818723 
382011 
885323 
388658 
3892016 
895899 
-3898807 
402239 
- 405696 


11.409180 
-412689 
416225 
419788 
423378 
426995 
480641 
-434316 
488020 


11,441753 








ae | | | ———_——_  ——_—_—___ | — 





AND EXTERNAL SECANTS 179 



























































88° 89° 

*| Vers. |D.1".] Ex.sec. |q+2/| ’ | Vers. |D.1".| Ex. sec. aise 
15, 29* L 15, 30* 

0 | 9.984573 | 2.17 } 11.441753 7 9086 |f 0 | 9.992854 | 2.18 | 11.750498 | 6801 
1 .984703 | 2.18 .445517 | 9215 1 . 992482 | 2.15 * .757925 6929 
2! (9848384 | 2.18 |  .449311 | 9345 || 2] .992611 | 2.18] 765477 | 7056 
3| 1984965 | 2.18 | - .453137 | 9474 || 3 | 1992730 | 2.15 | .773158 | 7184 
4) .985096 | 2.17]  .456994 | 9603 || 4 | .992868 | 2.13]  .780973 | 7312 
5 | .985226 |2.18 | .460883 | 9732 || 5 | .992996 | 2.13}  .788926 | 7440 
6| .985357 | 2.17 |  .464805 | 9862 || 6 | .993124 | 2.15 | .797022 | 7567 
7] .995487 | 2.18{ 468761 | 9991 || 7 | .993253 | 2.13 |  .805268 | 7695 
8] .985618 | 2.17 | .472751 |4120 || 8 | .993381 | 2.13] .813668 | 7823 
9+ 1985748 | 2.18] .476775 | 0249 || 9 | .993509| 2.13 | 822229 | 7950 
10 | .985879 | 2.17 | 480834 | 0378 ||-10 | .993637 | 2.13 | 880956 | 8078 
11 | 9.986009 | 2.18 | 11.484929 | o507 || 11 | 9.993765 | 2.15 | 11.839858 | 8205 
12 | .986140 | 2.17] .489061 | 0636 || 12 | .993804 | 2.13 | .848940 | 8333 
13 | .986270 | 2.17 | .493230 | 0765 || 13} .994022 | 2.1384  .858211 | 8460 
14] .986400 | 2.18 |  .497437 | 0804 || 14 | .994150 | 2.13 | 867679 | 8588 
15 | .986531 | 2.17] 501683 | 1028 || 15} .994278 | 2.13 |. .877351 | 8715 
16 | .986661 | 2.17 |  .505968 | 1152 || 16 | .994406 | 2.13 | 887289 | 8843 
17| .986791 | 2.17 | 510293 | 1281 || 17 | .994534 | 2.13 | .897350 | 897 
18 | .986921 | 2.17]  .514659 | 1410 || 18 | .994062 | 2.12 | :907697 | 9097 
19 | .987051 | 2.17 | — .519066 | 1589 | 19 | .994789 | 2.13 | 918290 | 9225 
20 | .987181 | 2.17]  .523516 | 1668 || 20] .994917 | 2.13 | 929141 | 9352 
21 | 9.987311 | 2.17 | 11.528010 | 1797 || 21 | 9.995045 | 2.13 | 11.940264 | 947 
22] 1987441 | 2.17 |  .532548 | 1925 || 22 | 995173 | 2.13 | 951672 | 9607 
23.| 987571) 2.17]  .587131 | 2054 || 23 | .995301 | 2.12 | 963381 | 9734 
24] 987701 | 2.17 |  .541760 | 2183 || 24 .995428 | 2.13 | .975408 | 9862 
25 | .987831 | 2.17 |  .546437 | 2312 || 25 | .995556 | 2.12 | 11.987769 | 9988 
26 | .987961 | 2.17]  .551164 | 2440 || 26 | .995683 | 2.13 | 12.000485 | #116 
27 | 988091 | 2.17 | .555935 | 2569 || 27] .995811 | 2.13] 013578 | 0243 
28 | 1998221 | 2.15 |  .560759 | 2698 || 28 | 995939 | 2.12 | 027069 | 0370 
29 | 1988350 | 2.17 | .565634 | 2826 || 29 | .996066 | 2.12}  .040984 | 0497 
30 | .988480 | 2.17 | .570561 | 2955 || 80} .996193 | 2.13 | .055352 | 0624 
31 | 9.988610 | 2.15 | 11.575542 | 3083 || 31 | 9.996321 | 2.12 | 12.070202 | o%5t 
32] 988739 | 2.17 |  .580578 | 3212 || 82 | .996448 | 2.13 | 085569 | 087 
33 . 988869 | 2.15 .585670 | 8340 || 33 . 99657 2.12 .101490 1005 
34| 1988998 | 2.17]  .590819 | 2469 || 84 | .996703 | 2.12 | 118008 | 1132 
35 | .989128 | 2.15 | .596027 | 3597 || 85 | .996830 | 2.12 |  .135168 | 1259 
36.| .989257 | 2.17 |  .601295 | 3726 || 86] .996957 | 2.13 | 158024 | 1386 
7 | .989387 | 2.15 |  .606625 | 3854 || 87 | .997085 | 2.12] 171634 | 1513 
38 | 989516 | 2.17 | 612018 | 3983 || 38 | .997212 | 2.12 | .191066 | 1640 
39 | 1989646 | 2.15 | .617475 | 4111 || 39 | .997339 | 2.12 | .211896 | 1767 
40} .989775 | 2.15 |  .622998 | 4239 || 40 | .997466 | 2.12 | .232712 | 1894 
41 | 9.989904 | 2.17 | 11.628589 | 4368 || 41 | 9.997593 | 2.12 | 12.255116 | 2020 
42] .990034 | 2.15 |  .634250 | 4496 || 42 | .997720 | 2.12 278723 | 2147 
43 | .990163 | 2.15 |  .639982 | 4624 || 43 | .997847 | 2.12 303674 | 2274 
44] .990292 | 2.15 |  .645788 | 4752 || 44 | 997974 | 2.12 830129 | 2401 
451 .990421 | 2.15 |  .651668 | 4881 || 45 | .998101 | 2.12 358285 | 2527 
46 | .990350 | 2.15 |  .657626 | 5009 || 46 | .998288 | 2.12 388875 | 2654 
7 | .990679 |.2.15 |  .663663 | 5137 || 47 | .998855 | 2.10 420686 | 2781 
48 | .990808 | 2.15 | 669781 | 5265 || 48 | .998481 | 2.12 455575 | 2907 
49 .990937 | 2.15 .675984 | 5393 || 49 . 998608. | 2.12 493490 3034 
50 | .991066 | 2.15 |  .682272 | 5521 || 50 | .998735 | 2.12 535009 | 38161 
B1 | 9.991195 | 2.15 | 11.688649 | 5649 || 51 | 9.998862 | 2.10 | 12.580893 | 8287 
52 | .991324 | 2.15 | 1695117 | 5777 || 52 | .998988 | 2.12] 682172 | 34i4 
53] .991453 | 2.15 | 701679 | 5905 || 53} .999115 | 2.10 |. .690291 | 3540 
54] .991582 | 2.13 |. .708838 | 6033 || 54} .999241 | 2.12] 757364 | 3667 
55 | .991710 | 2.15 | .715097 | 6161 |) 55 | .999368 | 2.10 | 836672 | 3793 
56 | .991839 | 2.15 | .'721958 | 6289 || 56 | .999494 | 2.12 | 12.933708 | 3920 
57 | .991968 | 2.13 | 728925 | 6417 || 57 | 1999621 | 2.10 | 13.058774 | 4046 
58 | .992096 | 2.15 |  .736002 | 6545 || 58 | .999747 | 2.12 | 234991 | 4172 
59 | 999225 | 2.15 | .743192 | 6673 || 59 | 999874 | 2.10 | _ 536148 | 4299 
60 | 9.992354 | 2.13 | 1.750498 | 6801 || 60 10.000000 | 2.10 Inf. pos. | 4426 
15.30* 15.31% 








180 


= 


SHIA BWW! 





| .01018} . 


| 01193} .§ 





0° 


Sine |Cosin 


1° 


Si ne Cosin 


TABLE XXVII.—NATURAL SINES AND 





9° 


Sine |Cosin 








3° 


Sine |Cosin 


COSINES 
Go 


Sine 


Cosin 





00000 
.00029 
00058 
00087 
.00116 
00145 
00175 
00204 
00233 
00262 
00291 


00320) . 
.00349 | .99999 
00878 | . 
00407}. 
00436 | . 
00465}. 
00495} . 
00524 | . 
00553 | . 
90582} . 


00611} . 
00640} . 
. 00669 | . 
00698 | . 
00727] . 
00756 | . 
00785 | . 
00814 |. 
00844 | . 
00873}. 


00902) . 
00931 |. 
.00960) . 
00989 | . 


One. 
One. 


.01047|. 
01076} . 
01105). 
01134}. 
01164) . 


01222) . 
.01251 | .99992 
01280}. 
.01309) . 
01338 |. 
01367}. 
01396 |. 
.01425}. 
01454. 


01483 |. 
01513} . 
-01542 | . 
01571). 
.01600) . 
01629}. 
.01658 
01687 |. 
01716). 
01745}. 


Cosin 

















.99985 
99984 
3| 99984 
99983 
99983 
. 99982 
99982 
99981 
99980 
. 99980 
.99979 
.99979 
.99978 
Boob, 
| 999. 
99976 
| .99976 
99975 
99974 
99974 
99973 


99972 
99972 
99971 
99970 
.99969 
.99969 
.99968 
- 99967 
. 99966 
.99966 


.99965 
99964 
. 99965 
.99963 
. 99962 
.99961 
.99960 
. 99959 
-99959 
99958 


.99957 
- 99956 
. 99955 
99954 
- 99953 
99952 
99952 
-99951 
99950 
99949 


. 99948 
.99947 
99946 
-99945 
.99944 
99943 
99942 
.99941 


02327 
.02356 
02385 
02414 
02443 
02472 
02501 
02530 
.02560 
02589 
02618 


-02647 
02676 
02705 
02734 
02763 
02792 
02821 
02850 
02879 
02908 


02938 
-02967 
02996 
03025 
03054 
03083 
.03112 
-03141 
03170 
.03199 


.03228 
03257 
03286 
03316 
03345 
03374 
03403 
03432 
.03461 
03490 


-99940)| . 
"99939 || . 


. 99939 
.99938 
99937 
99936 
3) 99935 
. 99934 
99933 
. 99982 
99951 
. 99930 
.99929 


99927 
- 99926 
99925 
99924 
99923 
99922 
99921 
=99919 
-99918 
99917 


99916 
99915 
.99918 
99912 
.99911 
.99910 
.99909 
.99907 
. 99906 
2|.99905 


99904 
. 99902 
.99901 
.99900 
99898 
99897 
. 99896 
.99894 
|.99893 
. 99892 


. 99890 
. 99889 
. 99888 
99886 
99885 
99883 
99882 
. 99881 
99879 
99878 


. 99876 
99875 
. 99873 
99872 
99870 
. 99869 
. 99867 
.99866 
.99864 
99863 














05284 | 99863 
05263 |. 99861 
05292 | .99860 
05321 | .99858 
05350 |. 99857 
05379 | .99855 
05408 | 99854 
05487 | 99852 
05466 | 99851 
05495 
05524 


05553 
05582 
05611 
-05640 
-05669 
05698 
05727 
05756 
05785 
-05814 


05844 
05878 
05902 
05931 
-05960 
05989 
.06018 
-06047 
06076 
.06105 


99847 


99846 | 
99844 
-99842 
- 99841 
99839 
99838 
99836 
99834 
99833 
99831 


. 99829 
99827 


99821 
.99819 
.99817 
.99815 
.99813 


.99810 


99806 
. 99804 
.99803 
.99801 


99797 
99795 
99793 
.99790 
99788 





99784 
-99780 
9977 
99774 
99770 | 


99766 
. 99764 


99760 
99758 
99756 





.99849}| . 


"99826 | . 
"99824 || « 
“99822 |. 





99812 . 
“99808 | . 


.99799 || . 


“99792 | . 





“99786 || . 
"99782 |: 


99776 || . 


99772). 
99768 ||. 


‘99762 | . 





06976 
.07005 
07034 
.07063 
.07092 
07121 
07150 
07179 
07208 
1237 
07266 
07295 
07324 
07353 
07382 
07411 
07440 
-07469 
07498 
07527 


07556 











- 99756 
99754 
99752 
99750 
99748 
99746 
99744 
.99742 
.99740 
.99738 
99736 
99734 
99731 
99729 
99727 
99725 
99723 
99721 
-99719 
99716 
99714 


99712 
.99710 


3|.99708 


99705 
.99708 
99701 
.99699 
. 99696 
99694 
-99692 


. 99689 
99687 
99685 
.99683 
. 99680 
.99678 
99676 
99673 
99671 
99668 


. 99666 


. 99664 | 


.99661 
.99659 
99657 
. 99654 
99652 
.99649 
99647 
99644 


5! .99642 


.99639 
.99637 
99635 
.99632 
99630 
99627 
-99625 
99622 
.99619 








Sine 


88° 


Cosin 








Cosin| Sine 


87° 








Cosin | Sine 








86° 


Sine 


85° 





le ove) 


lomwwnaad 


aa es 


TABLE XXVII.—NATURAL SINES AND COSINES 























181 



























































i tte 6° Mes he ee 9° ; 
Sine /Cosin |} Sine |Cosin || Sine |Cosin || Sine |Cosin || Sine |Cosin 
0 | .08716| .99619}} 10453) .99452}| .12187) .99255 || .18917 | .99027 || .15643| .98769! 60 
1 | .08745| .99617 || .10482! .99449 || .12216/ .99251 || .18946] .99023 || .15672 -98764 | 59 
2 | .08774| .99614|| .10511 | .99446 || . 12245) .99248 |) .18975) .99019)| .15701| .98760: 58 
8 | .08803} .99612)| .10540} .99443 || .12274| .99244/| .14004| .99015 || .15730 98755, 57 
4 | .08831} .99609 || . 10569} .99440 || .12302) .99240)| .14033} .99011 || .15758|.98751, 56 
5 | .08860] .99607 || .10597 | .99437 || .12831] .99237); .14061| .99006]| .15787| .98746 55 
6 | .08889! .99604}| .10626 | .99434 || 12360) .99233 || .14090| .99002 || .15816|.98741 54 
7% | 08918] .99602]| .10655| .99431 || .12389] .99230|} .14119| .98998 || .15845|.98737 53 
8 | .08947 99599 || .10684 . 99428 || .12418| .99226 || .14148] .98994|| .15873| .98732 52 
9 | .08976; .99596 || .10718 | .99424 || .12447) .99222|| .14177 | .98990)| .15902) .987 oF xh! 
10 | .09005 | .99594 |} .10742| .99421 |} .12476) .99219|| .14205 | .98986 || .15931 | .98723, 50 
- 11 | .09034! .99591 || .107'71| .99418 | .12504| .99215 || 14234] .98982]| .15959|.98718 49 
2 | .09063)| .99588)| . 10800} .99415 |} .12533| .99211 || .14263 | .98978 || .15988}.98714) 48 
13 | .09092) .99586 || . 10829} .99412 || .12562| .99208|| .14292] .98973 || .16017| .98709; 47 
14 | 09121) .99583}| .10858 | .99409 || .12591 | .99204/| .14320)| .98969 || .16046) .98704| 46 
15 | .09150} .99580}| .10887 | .99406 || .12620|.99200 || .14349| .98965 || .16074!|.98700) 45 
16 | .09179| .99578)| .10916| .99402 || .12649| .99197|| .14878] .98961 || .16103] .98695) 44 
47 | .09208| .99575|| .10945] .99399 || .12678].99193|| .14407} .98957 || .16132|.98690, 43 
18 | .09237 | .99572|| .10973| .99396 || .12706 | .99189 || .14436 | .98953 || .16160) 98686 42 
19 | .09266| .99570}| .11002) .99393 || .12735|.99186 || .14464| .98948|| .16189 .98681 | 41 
20 | .09295)| .99567)| .11031|.99390 || .12764] .99182 || .14498! 98944 || .16218].98676) 40 
21 | .09324| .99564|| .11060] .99386 || .12793| .99178 || .14522| .98940|| .16246 | .98671| 39 
22 | .09353| .99562}| .11089| .99383 |} .12822| .99175|| .14551| .98936 || .16275 | 98667 38 
23 | .09382) .99559}|| .11118] .99380 |} .12851|.99171|| .14580) .98931 || 16304) .98662, 37 
24 | .09411| 99556!) .11147| .99377|| .12880| .99167)! .14608| .98927 || .16333|.98657 | 36 
25 | .09440) .99553}| .11176|.99374|| .12908] .99163)| .14637| .98923 || .16361| .98652 35 
26 | .09469) .99551|| .11205|.99370 | .12937 | .99160|| .14666| .98919|| .16390 .98648 | 84 
27 | .09498| .99548]| .11234! .99367 || .12966 | .99156)} .14695/ .98914|| .16419| 98643) 33 
28 | .09527| .99545}| .11263) .99364!| .12995}| .99152)| .14723) .98910 .16447 | .98638) 32 
29 | .09556| .99542|| .11291] .99360 |} .13024] .99148)} .14752| .98906 || .16476| 98633) 31 
30 | .09585| 99540] | .11320| 99357 || .13053 | .99144|| .14781| .98902|| .16505| .98629) 30 
31 | .09614] .99537|| 11349] .99354'| .13081|.99141 || .14810) .98897|' .16533} .98624) 29 
32 | .09642] .99534|| .11378) .99351 || 18110] .99137 || .14838| 98893) . 16562) .98619 28 
33 | 09671) .99531|| .11407| .99347 || .13139 | 99133 || .14867! .98889|| .16591 | .98614| 27 
34 | .09700] .99528)| .11436| .99344'| .13168| .99129 || .14896| .98884|| .16620|.98609) 26 
35 | .09729) .99526) | .11465| .99341 || .13197| .99125 || .14925| .98880|) .16648| .98604) 25 
36 | .09758} .99523]| .11494| .99337 | .13226| .991221| .14954| .98876|| .16677 | .98600) 24 
37 | .09787| .99520|| .11523] .99334 | .13254] .99118}| .14982| .98871)| .16706| .98595 | 23 
38 | .09816| .99517|| .11552) .99331 || .13283] .99114|| .15011| .98867|| .16734| 98590) 22 
39 | .09845] .99514]| .11580] .99327 || .13312| .99110|| .15040| .98863 || .16763)| .98585 | 21 
40 | .09874| .99511}| .11609| .99324 || .13341| .99106|| .15069 | .98858 || .16792| .98580) 20 
41 | .09903} .99508}| .11638] .99320 || .13370| .99102/| .15097 | .98854 || .16820| .98575| 19 
42 | .09932| .99506| | .11667| .99317|| .13399| .99098 || .15126) .98849)| .16849| .98570| 18 
43 | .09961| .99503|| .11696| .99314)| .13427| .99094 || .15155 | .98845|| .16878) 98565) 17 
44 | .09990) .99500]| .11725|.99310 || .13456| .99091 || .15184| .98841 || .16906| .98561| 16 
45 | .10019| .99497}| .11754|+99307 | .13485 | .99087 || .15212| .98836 || .16935 | .98556) 15 
46 | .10048]| .99494|| .11'783] .99303 || .13514|.99083 || .15241| .98882]| .16964| .98551| 14 
47 | .10077| .99491|| 11812) .99300|| .13543 |.99079 || .15270| .98827|| .16992) .98546| 13 
48 | .10106} 99488] | .11840| .99297 || .13572 | .99075 || .15299 | .98823)| .17021| .98541| 12 
49 | .10135| .99485|| .11869 | .99293 || .13600 | 99071 || .15327'| .98818 || .17050 98536 11 
50 | .10164|.99482]| .11898|.99290 || .13629 | .99067 || .15356 | .98814|| .17078| 98531} 10 
51 | 10192! .99479|| .11927| .99286 || .13658 | .99063|| .15385 | .98809|| .17107| 98526, 9 
52 | 10221] .99476]| .11956| .99283|| .13687 | .99059 || .15414 | .98805 17186 .98521| 8 
53 | .10250] .99473]| .11985)| .99279 |) .13716 | .99055 || .15442| .98800)| .1 (164 .98516| 7 
54 | .10279| .99470|| .12014|.99276 || .13744|.99051 || .15471 | .98796|| .17198) .98511 6 
55 | 10308 .99467|| .12043| .99272 || .13773|.99047 || .15500| .98791 || 17222] 98506, 5 
56 | .10337|.99464|| .12071|.99269 || .13802|.99043 || .15529|.98787]|| .17250|.98501) 4 
57 | 10366) .99461}| .12100) .99265/| .13831|.99039 || .15557 | .98782 17279 .98496 3 
58 | .10395 | 99458] | .12129| .99262 || .13860/ .99035|| .15586 | .98778|| .17308|.98491) 2 
59 | 10424] .99455|| .12158 | .99258 || .18889|.99031 || .15615 | .98773 || .17336 .98486; 1 
60 | .10453 | .99452]| .12187| .99255 || .13917| .99027)| .15643 | .98769 || .17365 | 98481 0 
Cosin | Sine’| Cosin| Sine || Cosin | Sine || Cosin | Sine || Cosin | Sine 
84° 83° 82° 81° 80° 





i 


182 TABLE XXVII.—NATURAL SINES AND COSINES 
10° <a 12° 13° 14° 


Sine Cosin!| Sine Cosin | Sine |\Cosin || Sine |Cosin || Sine |Cosin 


.17365 | .98481 || .19081 | 98163 .20791 |.97815 || .22495 | .97437 || .24192| .97030| 60 
.17393} .98476}| .19109| .98157 || .20820] .97809 || .22523 | .97480 || .24220| .97028} 59 
17422] .98471 || .19188| .98152 || .20848 | 97803 || .22552 | .97424 || .24249] 97015) 58 
.17451| .98466 || .19167) .98146 || .20877 | .97797 || .22580) .97417 || .24277| .97008| 57 
.17479| .98461 || .19195} .98140 | .20905) .97791)| .22608| .97411 || .24305|.97001| 56 
17508} .98455 || .19224} .98135 || .20933) .97'784'| .22637 | .97404|| .24383 | .96994| 55 
.17537| .98450 || .19252} .98129 | .20962| .97778 || .22665 7398 || /24362|.96987| 54 
. 17565) .98445 || .19281 98124 20990 | .97772 || .22693 | .97391 || .24390| 96980) 53 
.17594] .98440}| .19809}.98118 || .21019| .97766 || .22722]| .97384/| .24418] 96973] 52. 
17623} .98435!| .19338].98112 | .21047| .97760)| .22750)| 97378 || .24446| 96966! 51 
10 | .17651 | .98430}| .19366].98107 || .21076| .97754)| 22778) .97371 || .24474|.96959| 50 


.17680 | 98425 || .19395|.98101 || .21104| .97'748 || .22807 | 97365 || .24503|.96952) 49 © 
17708 | .98420)| .19423 98096, 21182) 97742 | .22835 | 97358 || .24531|.96945!| 48 
13 | .17737|.98414|| .19452] 98090 | .21161].97735 | 22863 | 97351 || .24559|.96937| 47 
14 | .17'766 | .98409 || .19481} .98084 |} .21189] .97729 || .22892 | .97345 || .24587| .96930) 46 
15 | .17794| .98404 || .19509|.98079 | .21218}.97723 || .22920 | 97338) | .24615| .96923| 45 
16 | .17823|.98399|| .19538] 98073 | .21246] 97717 | .22948] 97331 || .24644| 96916! 44 
17 | .17852| .98394)| .19566] .98067 || .21275| .97711 || .22977 | 973825 || .24672] .96909| 43 
18 | .17880 | .98389}| .19595} .98061 |) .21303 | .97705 |) .23005 | .97318 || .24700} .96902| 42 
19 | .17909 | .98383|| .19623] .98056 || .21331 | .97698 || .23033 | .97311 || .24728| .96894| 41 
20 | .17937 | .98378 || .19652| .98050 || .21360| 97692 | .23062| .97304|| .24756| .96887| 40 


21 | .17966 | 98373 || .19680} .98044 || .21388] .97686 | .23090 | .97298 || .24784| .96880) 39 
22. | .17995 | .98368}| .19709|.98039 | .21417| .97680|| .23118 | .97291}|| .24813] .96873| 38 
23 | .18023 | .98362)| .19737 | .98033 | .21445) .97673)| .23146 | .97284)| .24841) .96866| 37 
24 | .18052 | .98357 || .19766| 98027 || .21474] .97667 || .23175 | .97278]| .24869| .96858| 36 
25 | .18081 | .98352]| .19794| .98021 || .21502) .97661 |} .23203 | .97271|| .24897) .96851 | 35 
26 | .18109 | .98347|| .19823].98016 | .21530).97655 || .23231 | .97264|| .24925| 96844) 34 
27 | .18138 | .98341]|] .19851|.98010 |) .21559) 97648 || .23260| .97257 || .24954| .96837| 33 
. 18166 | .98336 || .19880| .98004 | .21587) .97642 || .23288 | .97251 || .24982) .96829| 82 — 
29 | 18195 | 98331 || .19908}.97998 | .21616| .97636 | .23316 | .97244|| .25010| .96822| 31 
30 | .18224 | .98325)) .19937| 97992 || .21644) 97630 | .23345 | .97237 || .25088 | .96815| 30 


81 | .18252|.98320)) .19965| .97987 || .21672] .97623 | .23373 | .97230|| .25066 | .96807| 29 
82 | .18281|.98315 || .19994) 97981 || .21701| .97617 || .23401 | .97223)| .25094) 96800} 28 
33 | .18309 | .98310)| .20022).97975 || .21729| .97611 || .23429) .97217|| .25122].96793| 27 
34 | 18338 | .98304|| .20051).97969 | .21758) .97604 | .23458) .97210|| .25151| 96786) 26 
35 | .18367|.98299)) .20079}.97963 | .21786| .97598 || .23486 | .97203 || .25179|-.96778| 25 
36 | .18395 | .98294|) .20108|.97958 | .21814| 97592 || .238514|.97196 || .25207|.96771| 24 - 
87 | .18424|.98288 || .20136| .97952 || .21843/ .97585 || .23542].97189|| .25285).96764| 23 
88 | .18452 | .98283)| .20165|.97946 | .21871| .97579 || .23571 | .97182|| .25263| .96756| 22 
39 | .18481|.98277|| .20193| .97940 | .21899|'.97578 || .23599 | .97176|| .25291 | .96749| 21 
40 | .18509 | .98272 || .20222| 97934 || .21928) .97566 || .23627 | .97169)| .25820] .96742| 20 


41 | .18538 | .98267 || .20250/ 97928 | .21956) .97560 || .28656 | .97162|| .25348| .96734| 19 
42 | .18567 | .98261 || .20279) .97922 | .21985) 97553 || .23684| 97155 || .25376 | .96727| 18 
43 | .18595 | .98256 || .20307].97916 | .22018) .97547 || .28712|.97148)| .25404| .96719| 17 
44 | .18624| 98250 || .20336).97910 | .22041| .97541|| .23740] 97141 || .25432} .96712| 16 
45 | .18652 |.98245 || .20364|.97905 || .22070| .97534 || .23769 | .97134|| .25460] .96705| 15 
46 | .18681 | .98240)| .20393) .97899 | .22098 | .97528 || .23797| .97127 || .25488| 96697} 14 
47 | 18710) .98234)| .20421).97893 || .22126| .97521 || .23825] .97120|| .25516] .96690| 13 — 
.18738 | .98229 |) .20450 | .97887 || .22155| 97515 || .23853| 97113 || .25545| 96682) 12- 
49 | .18767| 98223 || .20478| .97881 || .22183] .97508)| .23882|.97106 || .25573| .96675 | 11 
BO | .18795 | 98218 || .20507| .97875 || .22212| 97502 | .23910|.97100|| .25601 | .96667) 10 


51 | .18824 | .98212)| .20535) .97869 || .22240 | .97496 || .23938 | .97093 || .25629 |.96660) - 9 
52 | 18852 .98207 || 20563) 97863 || .22268 | .97489 | .23966 | 97086 || .25657 | .96653 
53 | .18881 /.98201 || .20592) 97857 || .22297 | .97483 || .23995 | .97079 || .25685 | .96645 
54 | .18910/ 98196 || .20620) .97851 || .22325 | .97476 | .24023 | .97072 || .25718 | .96638 
55 | .18938 | .98190)) .20649| .97845 || .22353 | .97470 | .24051 | .97065 || .25741 |.96630 
56 | .18967 | 98185 || .20677| .97839 || .22382 | .97463)| .24079|.97058 || .25769 | .96623 

7 | ,18995 | .98179)) .20706| .97833 || .22410).97457 || .24108 | .97051 || .25798 | 96615 
58 | .19024|.98174|| .20734) .97827 || .22438 | .97450)| .24136 | .97044|| .25826 | .96608 
59 | .19052 | .98168|| .20763) .97821 || .22467 | .97444|| .24164|.97037 || .25854 | .96600 
60 | .19081 | .98163|| .20791| .97815 || .22495 | .97437 || .24192|.97030|| .25882 | .96593 


Cosin | Sine || Cosin | Sine ||Cosin | Sine || Cosin | Sine || Cosin! Sine 


79° 78° | 17 76° 75° 








= 





DHOIHOmwwmHO! 





bok fe 
wee 








2 
@ 











AN 
o"2) 
































~ lomwwnaasz20 


























TABLE XXVII.—NATURAL SINES AND COSINES 
19° 


Sine |Cosin 


15° 


Sine |Cosin 


~~ 


16° 


Sine 


Cosin 


17° 


Sine 


Cosin 





25882 96593 
.25910 | 96585 
25938 | 9657: 

.25966 | .96570 
25994! .96562 
26022) .96555 
26050} .96547 
.26079| .96540 
| 26107) . 96532 
26135) .96524 
26163) .96517 


.26191 | .96509 
12 | .26219| .96502 | 
13 | .26247| .96494 
14 | 26275} 96486 | 
15 | .26303) .96479 | 

~ 16 | 26331) .96471 | 

17 | .26359) 96463 | 
18 | 26387 |-. 96456 | 
19 | .26415| .96448 | 
20 | .26443) .96440 


21 | .26471| .96433 
2 | 26500) 96425. 
23 | 26528) 96417, 
24 | .26556| 96410 
25 | .26584| 96402 
26 | 26612] 96394 
27 | 26640! 96386. 
| 26668) 96379. 
| 26696 .96371 | 
80 | 26724) 96363 


31 | .26752) .96355 
32 | .26780) 96347 
- 383 | .26808) . 96340! 
34 | .26836) .96332 | 
35 | 26864) .96324 | 
36 | .26892) 96316) 
37 | .26920) .96308 
. 26948 | 96301 
. 26976 | 96293 
40 | .27004) .96285 


41 | 27032} .96277 
42 | .27060) .96269 
43 | .27088) .96261 
44 | .27116). 
45 | 27144! .96246 
46 | .27172) .96238 
47 | .27200| .96230 
48 | .27228] .96222) 
49 | 27256) 96214, 
50 | .27284) .96206 


51 | .27312) .96198 
52 | 27340) .96190. 
58 | .27368} .96182 
54 | 27396! 96174, 
dB | .27424| 96166 
56 | 27452) 96158! 
57 |.27480! .96150 
58 | 27508) .96142 
59 | .27536| 96134 | 
60 | 27564) .96126 


CORO RDM! 


10 


— 
—_ 


ww 
Noles) 


~) 


ee) 
oO 


S 
x 
oo 














27564 
27592 
27620 | 
27648 
27676 
27704 
27731 
27759 
27787 
27815 
27848 
27871 
27899 
27927 
27955 
27983 
28011 
. 28039 
28067 
28095 | 
28128 


. 28150 
28178 
. 28206 
28234 
28262 | 95923 
28290 
. 28318 
. 28346 
28374 
. 28402 


. 28429 
. 28457 
. 28485 
28513 
28541 
. 28569 
28597 
28625 
28652 
. 28680 


.28708 
. 28736 
28764 
28792 
28820 
. 28847 
28875 
. 28903 
. 28931 
.28959 


. 28987 
29015 
29042 
29070 
29098 
29126 
29154 
29182 
29209 
29237 


96126 
96118 
-96110 
96102 
96094. 
96086 
96078 
“96070 
96062 | 
96054 
96046 


96037 
96029 
96021 
96013 
96005 
95997 
95989 
95981 
95972 
95964 


95956 
95948 
. 95940 
95931 


95915 
95907 
. 95898 
.95890 
. 95882 


95874 
95865 
95857 
95849 
95841 
95832 
95824 
. 95816 
95807 
.95799 


95791 
95782 
95774 
95766 
95757 
95749 
.95740 
95732 
95724 
95715 


95707 || 


95698 
95690 
95681 
95673 
95664 
95656 
. 95647 
95639 
. 95630 


29237 
29265 
29293 
.29321 
29348 
29376 
29404 
29482 
29460 
29487 
29515 


29548 
29571 
29599 
29626 
29654 
29682 
29710 
29737 
29765 
29798 
29821 
29849 
29876 
29904 
29932 
29960 
29987 
80015 
30043 
30071 
30098 
30126 
30154 
30182 
80209 
80237 
80265 
30292 
30320 
30348 
80376 
30408 
30431 
30459 
30486 
30514 
30542 
8057 

80597 
£80625 


380658 
80680 
80708 
80736 
80763 
80791 
. 80819 
30846 
| 80874 
| .80902 


95630 | 


. 95622 


.95613 
95605 
95596 
. 95588 
95579 
95571 
. 95562 
95554 
95545 


95536 
95528 
95519 
95511 
95502 
95493 
95485 
95476 
. 95467 
95459 


. 95450 
95441 
95433 
95424 
95415 
95407 
95398 
95389 
95380 
95872 


95363 
95354 
95845 
95337 
95328 
95819 
. 95310 
95801 
95293 
95284 
-95275 
95266 
95257 
95248 
95240 
95281 


95222 


|. 95213 


95204 
95195 


95186 
(95177 
95168 
95159 | 
95150. 
95142 
95133 
95124 
95115. 
95106 


18° 


183 





Sine |Cosin 





. 80902 | .95106 
80929 | .95097 
80957 | .95088 
80985 | .95079 
.81012 | .95070 
.31040 | .95061 
31068 | .95052 


.31095 | .95043 


31123 | .95033 


.31151 | .95024 | 


-81178| .95015 


31206 | . 95006 
31233 | .94997 
31261 | .94988 
.81289) .94979 
31316 | .94970 
.81344)| .94961 
381872 | .94952 
. 31399 | .94943 
-31427 | .94933 
.31454 | .94924 


.31482 | .94915 
81510) .94906 
.81537) .94897 
.31565 | .94888 
.31593| 94878 
.381620| .94869 
.31648 | .94860 
.31675)| 94851 
.31703 | .94842 
381780 | .94882 


31758] .94828 
31786] .94814 
81813] .94805 
31841 | .94795 
31868] .94786 
81896 | .94777 
81923] .94768 
81951 | .94758 
81979-94749 
-82006| .94'740 
82034 | .94730 
32061 | .94721 
82089] .94712 
82116] .94702 
82144] .94693 
:32171 | .94684 
82199] 94674 
82227 | .94665 
82254 | 94656 
82282 94646 


.82309 | 94637 
82337 | 94627 
82364] . 94618 
82392 | .94609 
82419 | .94599 
82447 | .94590 
82474} .94580 
82502 | .94571 
82529 | ¢94561 
32557 | .94552 





82557 
382584 
382612 
82639 
382667 
82694 
82722 
82749 
82007 
82804 
82832 


382859 
82887 
82914 
82942 
32969 
82997 
83024 
383051 
383079 
33106 


88134 
38161 
83189 
33216 
33244 
388271 
83298 
383326 
383353 
383381 


. 83408 
. 83436 
-33463 
-33490 
38518 
33545 
83573 
33600 





.33627 
.33655 


38682 
383710 
88737 
383764 
83792 
33819 
83846 
83874 


33929 


. 83956 
33983 
384011 
384038 
384065 
34093 
84120 
84147 
84175 
384202 


88901 


94552 
94542 
94533 
94528 
94514 
94504 
94495 
94485 
94476 
94466 
94457 
94447 
94438 
94428 | 
94418 

94409 
.94399 
94390 


.94380 
94370 
. 94361 


94351) 
94342) © 


94332 


94322) : 
.94313| ¢ 


94303 
94293 
94284 
94274 
94264 


94254 | 
94245 


94285 
94225 
94215 
94206 
94196 
94186 


94176 | 


94167 


94157 
94147 
94137 
94127 
. 94118 
94108 
. 94098 
94088 


94078 


. 94068 


94058 
. 94049 
. 94039 
. 94029 
94019 
. 94009 
. 93999 
93989 
98979 
. 98969 








Cosin| Sine || Cosin 


73° 


nee 








74° 





Sine 








Cosin 





Sine 


72° 








Cosin| Sine 


(Se 


Cosin 











Sine 


70° 





lommowmoan 


184 


TABLE XXVII.—NATURAL SINES AND COSINES 





= 


cMIaonwoenel 


10 





Cosin 


21° 


Sine 


‘Cosin 


22° 


23° 


24° 





Sine Cosin 





Sine |Cosin 


Sine 


Cosin 





.93969 


29) 93959 


. 93949 
93939 
.93929 
.93919 
.93909 
. 93899 
93889 
98879 
5| .93869 


93859 
. 98849 
. 98839 
98829 


2.93819 


.93809 
. 93799 
.93789 
93779 
.93769 
.93759 
.938748 
93738 
93728 
.93718 
. 93708 
. 93698 
93688 
.93677 
.93667 
.9365? 
.93647 
. 93637 
. 93626 
.93616 


93606 | 


93596 
93585 


98575 | 


938565 


93555 | 


93544 
935384 
93524 
93514 
93503 
93493 


93483, 
93472 | . 
5} .93462, |. 


. 93452 
93441 


7) .93431 


93420 


. 93410 | 


-93400 
93389 


2} .93379 


93368 
93358 


35837 | . 9F 
35864 | . 9: 
35891 | . 
.385918 | . 
39945 | . 
.39973 | . 
.36000 | . 
. 36027 | . 
-36054 | . 
36081 | . 
.36108 | . 


86135 | . 
.36162} . 


.36190 
86217 
36244 
86271 
36298 
36325 
36352 
86379 


36406 | 


36434 
.36461 
.36488 
86515 
.86542 
. 386569 
. 36596 
. 36628 
. 36650 


86677 
86704 
.86731 

36758 
.386785 
36812 
86839 
386867 
36894 
36921 


36948 
386975 
37002 
81029 
37056 
37083 


3| 92762 





938222 
93211 
93201 
.93190 
.93180 
. 93169 
.93159 
.93148 


. 93137 
93127 
. 98116 
.93106 
93095 
93084 
93074 
. 93063 
93052 
98042 
93031 
93020 
. 93010 
92999 
92988 


92978 | 


92967 
92956 


92945 | 


929385 


92924 
92913 
92902 
92892 
92881 
92870 
. 92859 
92849 
- 92838 
92827 
. 92816 
92805 
92794 
92784 
927% 

92751 
92740 
92729 





87461 | .92718 
37488 | 92707 
37515 | .92697 
37542 | . 92686 
387569 | 92675 
.37595 | .92664 
87622 | .92653 


|| .87649| 92642 


387676 | 92631 
ts . 92620 


co 
ie 2) 
ae 
(or) 
_ 
= 
NS 
ce 
—) 


| 88295 | .9237'7 
38822 | . 92366 
38349 | 92355 
38376 | .92348 
88403 | 92332 
88430 | 92321 
88456 | .92310 
38483 | 92299 
88510 | 92287 
88537 | 92276 


88564 | .92265 
88591 | .92254 
88617 | 92243 
38644 | 92231 
88671 | .92220 
38698 | .92209 
80125 | .92198 
38752 | .92186 
38778 | 92175 
. 38805 | . 92164 


38832) .92152 
38859 .92141 
.38886 | 92130 
38912) .92119 
38939 | 92107 
38966-92096 
38993 | 92085 
39020 92073 
39046, 92062 





92718 


. 39073 | .92050 


389073 
39100 
.89127 
.389153 
.389180 
389207 
89284 
389260 
89287 
389314 
89341 


39367 
39394 
389421 
39448 
89474 
389501 
89528 
89555 
89581 
389608 


89635 


389688 
89715 
39741 
39768 
89795 
89822 
.39848 
389875 
39902 
39928 
.89955 
89982 
-40008 
-400385 
.40062 
-40088 
.40115 
40141 


.40168 
-40195 
40221 
40248 
40275 
-40301 
-40328 
40355 
40381 
40408 | 


.40484 
.40461 | 
-40488 | 
.40514 
40541 | 
40567 | 
.40594 
-40621 | 
.40647 
40674 | 





.39661 | .91 


92050 
92039 
92028 
92016 
92005 
.91994 
91982 
91971 
.91959 
.91948 
.91936 


91925 
.91914 
.91902 
91891 
91879 
.91868 
. 91856 ; 
91845 
91833 
91822 


.91810 
799 | 
91787 


91775 


91764 | 
91752 
91741 
91729 
91718 
91706 


. 91694 
. 91683 
91671 
.91660 
91648 
.91636 
91625 
.91613 
.91601 
.91590 


91578 
.91566 
91555 
91543 
.91531 
.91519 
.91508 
.91496 
91484 
91472 


91461 
.91449 
91437 
. 91425 
.91414 
.91402 


91390 
.91378 
- 91366 


91855 


40674 | 
40700 | 
40727 | 


40753 


40780 | 


-40806 
-40833 
.40860 
-40886 
.40918 
.40939 


.40966 
.40992 
-41019 
.41045 
.41072 
.41098 
-41125 
.41151 
.41178 
41204 


41231 
41257 
41284 
-41310 
41387 
41363 
-41390 
41416 
41443 
.41469 


.41496 
41522 
41549 
41575 
-41602 
.41628 
-41655 

41681 
41707 
41784 
.44760 
41787 
41813 
.41840 
41866 
41892 


41919). 


41945 


41972). 
41998) 


42024 
42051 
42077 
-42104 
-42130 
42156 
42183 
-42209 
42285 
42262 


v 


-91355 | 60 


91343 | 
ee 
.91319) 5 
91307 7| 
91295} 
.91283 | 
91272) 
91260 
91248, 
91236: 


91224 
91212. 
91200. 
91188 
91176 
91164 
91152 
“91140! 
91128 
91116 


91104 | 


91092. : 


91080 | 


91068 | ¢ 


91056 | 


91044! ; 


91032 | 
91020, 
91008, 


90996 | & 


. 90984 
90972 | 
90960) 
. 90948 
90936 
90924 
90911 
.90899 | 
90887 
90875 


. 90863 
90851 
. 90839 
90826 
. 90814 
. 90802 
90790 
9077 
90766 
90753 
90741 
90729 
90717 
90704 
.90692 
. 90680 
. 90668 
90655 
. 90643 
. 90631 











Sine 


69° 








Sine 


68° 





Cosin | Sine 


67° 











Cosin | Sine 








Cosin 





Sine 


fe 


6 
55 
54 
53 
52 
51 
50 


49 
48 
47 
46 


45 
44 
43 
42 
41 
40 
39 
38 
37 
36 
35 
34 
33 
32 
31 
30 
29 
28 
27 
26 
25 
24 
23 
22 
21 
20 
19 
18 
17 
16 
15 
14 
13 
12 
11 
10 

9 


lomwcmorcn-200 


a 


TABLE XXVII.—NATURAL SINES AND COSINES 





25° 


~ 


Sine 


Cosin 


26° | 


Sine |Cosin 


185 





27° 


Sine 


Cosin 


28° 


Sine 


Cosin 


29° 


Sine 


Cosin 





42262 
42288 
42315 
42341 
42367 
42394 
42420 
42446 
42473 
"42499 
42525 


11 | .42552 
12 | .42578 
13 | .42604 
14 | .42631 
15 | .42657 
16 | .42683 
17 | .42709 
18 | 42786 
19 | .42762 
20 | .42788 


21 | .42815 
22 | .42841 
23 | .42867 
24 | .42894 
25 | .42920 


i 
Somuomnewnwnol 





90631 
.90618 
.90606 
90594 
. 90582 
. 90569 
90557 
90545 
90582 
. 90520 
90507 


.90495 
90483 
90470 
90458 
. 90446 
.90433 
. 90421 
.90408 
.90396 
. 90383 


90371 
90358 
90346 
90334 
90321 
-90309 
. 90296 
. 90284 
90271 
90259 


90246 
. 90233 
90221 


5| .90208 


.90196 
90183 
90171 
90158 
.90146 
90133 


90120 
.90108 


2} .90095 


. 90082 
90070 
90057 
90045 
90032 


90019 | 


90007 


| 89994 


.89981 
.89968 
89956 
89943 
.89930 
.89918 
89905 
89892 
89879 





43837 
43863 
.43889 
.43916 
.43942 
.43968 
.43994 


89879 
89867 
89854 
89841 
.89828 
89816 
.89803 
89790 
89777 
2) .89764 
89752 


89739 
89726 
.89718 
.89700 
.89687 
89674 
89662 
89649 

















.45399 
-45425 
45451 
45477 
.45503 
-45529 
.45554 
45580 
-45606 
45632 
45658 


-45684 
.45710 
.45736 
.45762 
45787 
.45813 
.45839 
.45865 
45891 
-45917 


|| .45942 


.45968 
.45994 
-46020 
.46046 
46072 
.46097 
.46123 
.46149 


3}| 46175 


.46201 |. 
-46226 | . 
-46252 |. 
46278 | . 
.46304 |. 
.46330}. 
46355 | . 
.46381 |. 
.46407 |. 
46483 | . 


-46458 | . 
.46484 |. 
-46510) . 
-46536 | . 
46561 | . 
.46587 |. 
.46613 |. 
.46639 | . 
.46664 | . 
-46690 |. 








89101 
.89087 
89074 
89061 
.89048 
89035 
89021 
.89008 
88995 
.88981 
88968 


88955 
.88942 
88928 
88915 
.88902 
. 88888 
88875 
88862 
.88848 
88835 


88822 
88808 
.88795 
88782 
88768 
88755 
88741 
88728 
88715 
88701 











46947 
.46973 
-46999 
47024 
-47050 
47076 
47101 
47127 
47153 
47178 
47204 
47229 
47255 
47281 
47306 
-47382 
47358 
47383 
.47409 
47484 
.47460 
47486 
47511 
47537 
47562 
47588 
47614 
47689 
.47665 
47690 
47716 
47741 
47767 
47793 
47818 
47844 
47869 
47895 
47920 
-47946 
47971 


47997 
.48022 
.48048 
.48073 
.48099 
48124 
.48150 
48175 
48201 
48226 


48252 


48277 |. 


48303 


48328 |. 


48354 
.48379 
-48405 
.48430 
48456 
.48481 


88295 
.88281 
88267 
88254 
.88240 
88226 
88213 
.88199 
88185 
.88172 
.88158 
88144 
88180 
88117 
88103 
88089 
88075 
88062 
.88048 
88034 
. 88020 


.88006 
87993 
87979 
87965 
87951 
87937 
87923 
87909 
.87896 
87882 


87868 
87854 
87840 
87826 
87812 
87798 
87784 
87770 
87756 
87743 
87729 
87715 
87701 
87687 
87673 
87659 
87645 
87631 
87617 
87603 
87589 


8757 

87561 
7546 
87532 
87518 
87504 
.87490 
87476 
87462 


.48481 
48506 
-48582 
48557 
-48583 
.48608 
.48634 
-48659 
48684 
.48710 
48735 
48761 
48786 
48811 
48837 
.48862 
48888 
.48913 
48938 
48964 
.48989 


49014 
49040 
.49065 
.49090 
.49116 
.49141 
.49166 
-49192 
49217 
49242 


49268 
49298 
49318 
.49344 
.49369 
.49394 
.49419 
-49445 
.49470 
49495 


49521 
-49546 
49571 
-49596 
.49622 
.49647 
.49672 
.49697 
49728 
.49748 


49773 
.49798 
49824 
.49849 
49874 
.49899 
49924 
.49950 
.49975 
.50000 








87462 
87448 
87434 
87420 
87406 
87391 
87377 
. 87363 
87349 
87335 
87321 


87306 
87292 
87278 
87264 
87250 
87235 
87221 
87207 
87193 
87178 


.87164 
87150 


87136 | ¢ 


87121 
87107 
87093 
87079 
87064 
87050 
87036 


87021 
87007 
86993 
.86978 
86964 
.86949 
86935 
86921 
86906 
86892 


86878 
.86863 
.86849 
86834 
.86820 
86805 
86791 
86777 
.86762 
86748 


86733 


86719 


86704 
.86690 
86675 
.86661 
.86646 
.86632 
86617 
86603 











Sine 





























Cosin 





Sine 








Cosin 


Sine 


60° 








lommamaa2we 


186 


Dowco rmmol 








30° 


Sine |Cosin 


.50000 
50025 
. 50050 
50076 
.50101 
.50126 
50151 
50176 
50201 
50227 
50252 


51504 





.86603 
86588 
86573 
86559 
86544 
86530 
86515 
86501 
.86486 
86471 
86457 


86442 
2 | 86427 
86413 
86398 
86384 
3 | .86369 
86354 
.86340 
86825 
.86310 


86295 
3) .86281 
.86266 
3 | 86251 
.86237 
86222 
86207 
.86192 
.86178 
.86163 


.86148 
.86133 
.86119 
.86104 
.86089 
.86074 
86059 
86045 
.86030 
86015 


. 86000 
85985 
.85970 
85956 
85941 
85926 
.85911 
85896 
85881 
. 85866 


85851 
4) 85836 
85821 
85806 
85792 
85777 
85762 
85747 
.85732 
85717 


TABLE XXVII.—NATURAL 
31° 


Sine |Cosin 


82° 


Sine |Cosin 


SINES 





| 33° 


Sine 


AND 





Cosin 


COSINES 


34° 


Sine 


=—| 9 
‘Cosin 





51504 
-51529 
.51554 
.51579 
.51604 
51628 
.51653 
.51678 
-51708 
51728 
.51753 
51778 
-51803 
51828 
.51852 
51877 
-51902 
51927 
.51952 
51977 
.52002 


52026 
52051 





52745 


52770 
52794 
52819 
52844 
-92869 
52893 
.52918 
52943 
-52967 
52992 








52076 |. 
52101), 
| 52126. 
2151]. 
52175]. 
52200}. 
2225]. 
52250. 
52275]. 
52299] . 
52324 |. 
52349). 
52374. 
52399]. 
52423] . 
52448 |. 
52473 |. 
52498 | . 


52522). 
52547. 
52572). 
.52597 |. 
52621). 
-52646 |. 
52671]. 
52696 | . 
92720 | .8: 





85717 
85702 
85687 
85672 
85657 
85642 
85627 
85612 
85597 
85582 
85567 


85551 
855386 
85521 
85506 
.85491 
85476 
85461 
85446 
85431 
85416 


85401 
85385 


84959 


84943 
.84928 
.84913 
84897 
84882 
84866 
84851 
84836 
84820 
84805 





52992 | .84805 
.53017 | .84789 
538041 | 8477 


58066) .84759 | 


.53091 | .84743 


53115 | 84728 | 
.53140} .84712 | 


.53164| .84697 
.58189 | .84681 
53214] .84666 


538288) .84650 | 


53263! .84635 | 
.53288 | 84619. 


£58312) .84604 
53387 | 84588 
53861 |. 8457 


53386 | 84557 | 


53411 | 84542 


53485 | 84526 | 


.58460) .84511 
53484 | 84495 


53509 | 84480. 
58534 | 84464 | 


53558 | 84448 


.53583 | 84433 | 
.53607 | .84417 | 
53632 | .84402 | 
.53656 | £84386 | 
.53681 | 84370 


53705 | 84355 


53730 | 84339, 


58754 | 84324 
53779 | 84308 
58804} 84292 


"53828 | 84277. 


58853 | .84261 
53877 | 84245 


53902 | .84230 | 
53926 | 84214 | 


.53951 | :84198 
53975 | 84182 


. 54000 | 84167 
54024 | 84151 
.54049 | 84135 
54073 | 84120 
.54097 | 84104 
.54122 1.84088 
54146 | .84072 
54171 | 84057 
.54195 | .84041 
54220) .84025 


54244! .84009 
54269 | .83994 
154293 | .88978 
54317 | .83962 
54342 | .83946 
.54366 | .88930 
54391 | 83915 
.54415 | .83899 
54440 | .83883 
54464 | .83867 





| 54464 
54488 
54513 
54587 
| 54561 
.54586 
.54610 
54685 
54659 
54683 
54708 


54732 
54756 
54781 
54805 
‘| 54829 
|| .54854 
|| 54878 
54902 
54927 
54951 


54975 





.55678 
.55702 
55726 
55750 
5577 

.55799 
.55823 
55847 
55871 
55895 
55919 


{54999 |. 
‘| 55024 |, 

"55048 | . 8% 
55072). 
"55097 |. 
“55121 |. 
55145 |. 
155169 | . 
55194 . 


55218}. 
55242 |. 
55266 | . 
55291 | . 
|| .55315}. 
|| 55339 | . 
55363 | . 
.55388 | . 
|| .55412]. 
|| .55486 | . 8 


55460). 
55484 |. 
55509 | . 
.55588 | . 
.55557 | . 
.55581) . 
.55605 | . 
.55630). 
55654 | . 


-83867 
83851 
83835 
83819 
.838804 
.83788 
83772 
88756 
83740 
83724 
838708 
.83692 
83676 
.83660 
88645 
83629 
.88618 
88597 
83581 
838565 
83549 


83533 





83050 | 


83034 || . 


88017 
83001 


82985 4 


82969 
82953 
82936 
82920 
82904 





Cosin 


Sine 


59° 





Cosin 


Sine 


58° 





Cosin | Sine 


57° 


Cosin 








56° 


Sine 








83066 | . 











.55919 
55943 


.55968 | . 
55992 | . 
.56016 |. 
-56040 |. 
-56064 |. 
.56088 |. 
.96112}. 


.56136 
.56160 


56184 
56208 
56282 
56256 
56280 
56305 
56329 
56353 
56877 
.56401 


56425 
56449 
56473 
56497 
.56521 
.56545 
.56569 
.56593 


|| .56617 


56641 


56665 
.56689 
56713 
.56736 


.56760) . 
56784 |. 
56808 | . 
56882 |. 
56856 | . 
.56880 | .8: 


56904 |. 
56928 |. 
56952 . 
56976 |. 
-57000 |. 
57024 |. 
57047 |. 
|| 570711. 
“57095 | .82 
7119}, 


57143}. 
7167). 
.57191}. 
57215). 
57238 | . 
57262]. 
57286 | . 
.57310). 
57334}. 


.57358 
Cosin 





82904 60 
.82887 59 


82757 51 
82741 50 


‘s2724 


“82708 48 
82692 47 
82675 46 
82659 45 
82643 44 





82626 43 
82610 42 
(82593 41 
82577, 40 


82561 39 
82544 38 
82528, 37 
'82511 36 
(82495 35 
'82478 84 
82462. 33 
82446 32 
82429 81 
82413 80 


82396 29 
[82380 28. 
82363, 27 
'82347 | 26 











55°C 





TABLE XXVII.—NATURAL SINES AND COSINES 187 
35° 36° 37° 38° 39° 
Sine Cosin || Sine |Cosin|| Sine Cosin || Sine |Cosin || Sine |Cosin 
. 57858) .81915 || .58779 | 80902 || .60182 | .79864 || .61566 | .78801 || .62932!.77715| 60 
-57381 | .81899 || .58802 | 80885 || .60205 | .79846 || .61589 |. 78783 || .62955).77696| 59 
| 57405 | .81882!| .58826 | 80867 | .60228 | .79829 || .61612)|.78765 || .62977 | .77678 | 58 
-57429| .81865 || .58849 | 80850 | .60251 | .79811 || .61635|.78747 || .63000| .77660) 57 
| 57453 | 81848) | .58873 | 80833 | .60274 | .79793 || .61658| .78729 || .63022].77641| 56 
57477 | 81832) | .58896 | .80816 | .60298) .79776 | .61681 | .78711 || .63045| .77623| 55 
. 57501 | .81815 || 58920) .80799 | .60321 | .79758 || .61704 | .78694 || .63068}.77605| 54 
.57524| £81798 || 58943 | 80782 || .60344|.79741 || .61726 | .78676 || .63090|.77586| 53 
.57548 | 81782 || 58967) .80765 || .60367 | .79723 || .61749| .78658 | .63113}.77568| 52 
| 57572 | 81765) | .58990 | 80748 | .60390 | .79706 || .61772| .78640 || .63135|.77550) 51 
| 57596 | 81748) | .59014| 80730 | .60414 | .79688 || .61795 | .78622 || .63158] .77531| 50 


11 | .57619|.81731)| .59037|.80713 || .60487'| .79671|| .61818| .78604 || .63180|.77513| 49 
.57643 | 81714 || 59061 | 80696 || .60460 | .'79653 || .61841 |.78586 || .63203!.77494| 48 
13 .57667) .81698) | .59084| 80679 | .60483 | .79635 || .61864 | .78568 || .63225 .77476) 47 
14 | 57691) .81681|) .59108| .80662)| .60506| .79618 | .61887) .78550 || .63248 |. 
15 | .57715| .81664|| .59131 | 80644 || .60529| .79600 || .61909 .78532 || .63271 |. 
16 | 57738) .81647 || .59154| 80627 || .60553 | .79583 || .61982 | .78514/| .63293 | .'77421| 44 
17 | 57762) .81631 || .59178| 80610 | .60576 | .79565 || .61955 | .78496 || 63816 .7'7402) 43 
18 | 57786) .81614)| .59201 | .80593 | .60599| .79547 || .61978 | .78478)| .68838 .77384) 42 
19 | .57810) .81597|| .59225|.80576 | .60622).79530 | .62001 | .78460 || 63361). 
20 | .57833|.81580)| .59248/.80558 | .60645|.79512|) .62024|.78442|| 63383 | 77847) 40 


21 | 57857} .81563|| 59272) .80541 | .60668 | 79494} .62046! .78424!! .63406) .77329) 39 
22 | .57881| 81546 || 59295} .80524 | .60691| .79477 || .62069 | .78405 || .68428) .77310) 38 
23 | .57904) .81530)| .59818 | .80507 || .60714 | .79459 |) .62092| .78387 || .68451| .77292) 37 
24 | .57928) .81513|| .59342|.80489 | .60738 | .79441 || .62115|.78369 || 63473) .77273 36 
25 | .57952) .81496 || .59365}.80472 | .60761 | .'79424|) .62138) .78351 || .63496 | ..77255 | 35 
26 | 57976) 81479 || .59389 | 80455 | .60784| .79406 || .62160 | .78333|| .68518) 77286 34 
27 | .57999| .81462 || .59412| 80438 | .60807| .79388 || .62183 | .78815|| .63540) 77218 33 
28 | .58023| .81445 || .59436 | .80420 | .60830 | .79371)| .62206 | .78297 || .63563).77199| 32 
29 | .58047| 81428 || 59459} 80403 | 60853 | .79353 || .62229) .78279 || 63585) .7'7181| 31 
380 | .58070)| .81412)| 59482 | 80386 | .60876 | .79335 || .62251 | .78261 || .68608) .77162) 30 


31 | .58094| 81395 || .59506 | 80368 | .60899 | .79318)| .62274 | 78243! 636380) .7’7144| 29 
82 | .58118| 81378 || .59529 | .80351 |! .60922| .79300 || .62297 | .78225 || .63653| .77125 | 28 
83 | .58141| 81361 |) .59552 | .80334)| .60945| 79282}; .623820).78206 || .68675 | .77107 | 27 
34 | 58165} .81344 || .59576| 80316 || .60968 | 79264 | 62342) .78188|| 63698) .77088) 26 
85 | .58189) 81327 || 59599 .80299 || .60991 | .79247)| .62365|.78170)| .63720] .77070| 25 
86 | 58212) .81310)| .59622| 80282 || .61015| .79229 || .62388) .78152 || 63742) .7'7051| 24_ 
87 | 58236} .81293|| .59646 | .80264/| .61038 | .79211|| .62411 | .78184|| .63765) .77033| 23 
38 | 58260) .81276 || .59669 | .80247 |) .61061| .79193)| .62433|.78116 || .63787| .77014| 22 
89 | .58283| .81259)) .59693 | .80230)| .61084|.79176 | .62456|.78098 | .63810) .76996 | 21 
40 | .58307| .81242|| .59716 | .80212 || .61107| .79158 | .62479| . 78079 || .63832) .76977 | 20 


41 | 58330) .81225!| .59739 | 80195 || .61130) .79140 | .62502! 78061 || .63854! .76959| 19 
42 | .58354) 81208 || .59763) .80178)| .61153) .79122)| .62524| .78043 | .63877|.76940| 18 
43 | 583878 .81191 || .59786| .80160)| .61176 | .79105)|| .62547|.78025 | .63899| .76921| 17 
44 .58401) 81174) .59809 | 80143 | .61199 | .79087)) .62570 | 78007 || .63922|.76903 | 16 
45 | 58425 .81157|| .59832| 80125 || .61222) .79069 || 62592) .7’7988 | .63944| 76884) 15 
46 | 58449, 81140 | .59856 | .80108 | .61245 | .79051 || .62615 | .77970 | .63966 | .76866| 14 
1 | .58472) 81123 || .59879 | 80091 || .61268 | .79033)) .62638 | .77952 | .63989| .76847| 13 
48 | .58496|.81106 | .59902).80073 | .61291!.79016|| .62660) .77934 || .64011| .76828| 12 
49 | 58519) 81089 || .59926) .80056 | .61314|.78998  .62683|.77916 | .64033|.76810) 11 








~ 








a 





i 
10 


33 
ee 
co Or 
€© OO 
ee 
Ove 


~J 
=? 
(se) 
for) 
oO 
_— 
are 






































50 | .58543).81072 || .59949) .80038 || .61337! .78980 || .62706|.77897 || .64056| .76791| 10 
51 | .58567} .81055 || .59972) .80021 || .61360).78962 | .62728|.77879!| .64078|.76772| 9 
52 | 58590} .81038 || .59995|) 80003 | .61883|.78944) .62751|.77861 || .64100|.76754) 8 
53 | 58614) .81021 || 60019) .79986 || .61406 | .78926| .62774|.77843 || .64123) 76735 
54 | .58637| .81004 || .60042|.79968 | .61429|.78908| .62796|.7'7824 || .64145 |. 76717 

55 .58661 | .80987 || .60065|.79951 || .61451| .78891 |! .62819|.7'7806 || .64167 | .76698 
56 | .58684| .80970 || .60089|.79934 | .61474|.78873)/ .62842]| .77788 || .64190| .76679 

” | 58708} .80953 || .60112|.79916 | .61497] .78855|! .62864| .7'7769 64212 |. 76661 | 
58 | .58731|.80936 || .60135|.79899 | .61520}.78837)! .62887| .77751 64284 |. 76642 
59 | .58755 | .80919 || .60158|.79881 | .61543|.78819|| .62909| .77733 || 64256 .76623 

60 | .58779| 80902 || .60182) .79864 | .61566|.78801)) .62932|.77715 64279 .76604 

Cosin | Sine || Cosin| Sine || Cosin| Sine | Cosin| Sine || Cosin | Sine 


~ lommwop ore-2 




















54° 53° | 62° 51° 50° 


188 TABLE XXVII.—NATURAL SINES AND COSINES 
| a nner eee 
40° 41° 42° 43° 44° 
Sine |Cosin |} Sine |Cosin || Sine |Cosin Sine 'Cosin |} Sine |Cosin 


64279 | .76604 || .65606 | .75471 || .66913| .74314 || .68200 | .73135 || .69466 | .71934) 60 
64301 | .76586 || .65628 | .75452 || .66935] .74295 || .68221 | .73116 || .69487 | .71914) 59 
64323 | .76567 || .65650|.75433 || .66956| .74276 || .68242 | .73096 || .69508 | .71894| 58 
64346 | .76548 || .65672 |.75414 || .66978] .74256 || .68264 | .73076 || .69529 | .71873) 57 
64368! .76530 || .65694 |.75395 || .66999) .74237 || .68285 | .'73056 || .69549 | .71853) 56 
64390] .76511 || .65716 | .75375 || .67021| .74217 || .68306 | .73036 || .69570 | .71833) 55 
64412] 76492 || 65788 | 75356 || .67043) .'74198}| .68327 | .'73016 || .69591 | .71813) 54 
64435 | .76473 || .65759 | .'75337 || .67064| .74178 || .68349 | .72996 || .69612) .71792) 53 
644571. 76455 || .65781 |.'75318 || .67086) .'74159}| .68370| .'72976 || .69633 | .71772) 52 
.64479 | .76436 || .65803 | .75299 || .67107| .74139 || .68391 | 72957 || .69654) .71752) 51 
64501 | .76417 || .65825 | .75280 || .67129) .74120]| .68412 | .72987 || .69675 | .71732) 50 


11 | 64524! .76398 || .65847 | .75261 || .67151) .'74100]| .68434) .72917 || .69696| .71711) 49 
12 | 64546) .76380}| .65869 | .75241 || .67172/| .74080]| .68455 | 72897 || .69717 | .71691) 48 
13 | 64568 .76361 || .65891 | .75222 || .67194| .74061 || .68476 | .72877 || .69737 | .71671| 47 
14 | 64590! 76342 )| .65913].75203 || .67215) .74041 || .68497 | .72857 || .69758 | .71650) 46 
15 | .64612) .76323}| .65935 | .75184 |) .67237] .74022]| .68518| .72837 || .69779| .71630) 45 
16 | .64635) .76304)|) .65956 | .75165 || .67258] .74002|| .68539) .72817 || .69800 | .71610| 44 
17 | .64657 | .76286 || .65978 | .75146 || .67280] .73983]| .68561 | .72797 || .69821 | .71590| 43 
18 | .64679| 76267 || .66000 | 75126 || .67301] .73963|| .68582} .72777 || .69842|.71569) 42 
19 | .64701| .76248 | .66022).75107 || .67323) .73944)| .68608) .72757 || .69862) .71549) 41 
20 | .64723| .76229 | .66044 | .75088 || .67344| .73924 |) .68624 | .72737 || .69883|.71529) 40 


21 | 64746! .76210'| .66066 | .75069 || .67866) .73904 || .68645] 72717 || .69904|.71508) 39 
22 64768) .76192 | .66088 | .75050 || .67387] .'73885 || .68666 | 72697 | .69925) .71488) 38 
23 | 64790) .76173 || .66109 |.75030 | 67409} . 73865 || .68688 | .72677 || .69946 | .71468) 37 
24 | .64812).76154 || .66131|.75011 || .67430] .73846)| .68709| .72657 || .69966| . 71447) 36 
25 | 64834) .76135 || .66153 |. 74992 || .67452) .78826 || 68730) .72637 | .69987 | .71427) 35 
26 | 64856) .76116 || .66175|.74973 | .67473} ..73806)| .68751 | .72617 | . 70008) .71407| 34 
27 | 64878) .76097 || .66197 | .74953 | 67495] .'78787]| .68772] 72597 || 70029) .713886) 33 
28 | 64901) .76078 | .66218 | .74934 || .67516) 73767 || .68793| .72577 || .70049| .71366) 32 
29 | 64923) .76059 || .66240].74915 || .67538] .73747]| .68814] .72557 | .70070) . 71845) 31 
30 64945) .76041 || .66262 | 74896 || .67559) . 738728 || .688385 | .72587 || .70091 | .71825) 30 


31 | 64967) 76022, 66284 | 74876 || .67580] .'73708 || .68857 | .72517 || .70112| .71805) 29 
| 64989) .76003 || .66306 | .74857 || .67602| .73688 || .68878 | .72497 | 701382] .71284) 28 

Uy 4)| 66327 | .74838 | .67623! .'73669 || .68899 | 72477 || .70153 | .71264)| 27 
34 | .65033) 75965 || .66349|.74818 | .67645| .73649|| .68920| .72457 || .70174 | 71248) 26 
35 | .65055 | .75946 || .66371|.74799 | .67666| .'73629|| .68941 | 72437 || .70195 | .71223) 25 
36 | .65077 | .75927 || .66393 |.74780 | .67688] .'73610]| .68962} .72417 | 70215) . 71203) 24 
37 | 65100) .75908 | .66414|.74760 | .67'709) .73590 || 68983] .72397 || 70236) .71182) 23 
: ; 66436 |.74741 | .67'730] 73570|| .69004] 723877 || .70257| :71162) 22 
39 | .65144| .75870 || .66458 | .74722 | .67752) .'73551 || .69025 | . 723857 |). 70277 | 71141) 21 
40 | 65166) .75851|| .66480| .74703 | .67773) .73531 || .69046| . 72337 |) .70298).71121| 20 


41 | 65188) .75832 | .66501 74683, 67795 | .73511 || .69067 | .72317 || .70319) .71100) 19 


42 | 65210) .75813 | .66523 | .74664 || .67816| .73491 || .69088| .72297 || 70389) .71080) 18 
43 | .65282)|.75794 || .66545 | .74644 || 67837) .73472 || .69109 | .72277 || .'70360) .71059) 17 
44 | 65254) .75775 | .66566 | .74625 | .67859} .73452)| .69130] .72257 || . 70381) .71039) 16 
45 | 65276 .75756 || .66588 | .74606 | .67880) .73432|| .69151 | .72236)| .70401| .71019) 15 
46 | 65298) .75738 | .66610|.74586 | .67901 | .73413)| .69172] .72216|| ..70422) 70998) 14 
47 | 65320) .75719 | .66632 | .74567 || .67923| .73393 |) .69193| .72196 || .70443 | 70978) 138 
48 | 65342). 75700 | .66653 | .74548 | .67944| .73373 || .69214|.72176|| .70463| .70957| 12 
49 | .65364| .75680 | .66675 | .74528 | .67965 | .73353 || .69235}.72156)| .'70484| .709387) 11 
50 | .65386 | .75661 || .66697 | .74509 | .67987 | .73333 || .69256 | .72136 || .70505 |. 70916, 10 


51 | .65408| .75642|| .66718 | .74489 || .68008|.73314|| .69277| 72116 || .'70525! .70896 
52 | .65480) .'75623 || .66740 | .74470 || .68029) .'73294|) .69298 | ..72095 |) .70546| . 70875 
53 | .65452) .75604 || .66762 | .74451 | .68051 | .78274 || .69319 | .72075 || .70567) .70855 
54 | 65474! 75585 || .66783 | .'74431 || :68072] .'73254]| .69340] .72055 |; .'70587 | .'70884 
: 75566 || .66805 | .74412.|| .68093 | .73234 || .69361 | .72035 || 70608} .70813 
56 | .65518) .75547 || .66827 |.'74892 || .68115) .732151| .69382| .72015 || .'70628 | .'70793 
57 | .65540) .75528 | .66848 .74373 | .68136 | .73195 || .69403 | .71995 || .70649) . 70772 
58 | .65562| .75509 || .66870 | .74352 || .68157)| .73175]| .69424| .719741| .70670) . 70752 
59 | .65584) .75490 || .66891 | .74334 |) .68179|.73155|| .69445| .'71954/| .70690 | .70731 
60 | 65606) .75471 | .6@913) .74314 | .68200 | .73135 || .69466| 71934 | .70711 70711 

Cusin | Sine Cosin ! Sine || Cosin| Sine || Cosin} Sine || Cosin| Sine 
, Pe a ar ek —— 


49° | 48° | 47° 46° 45° 


» 
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TABLE XXVIII.—NATURAL TANGENTS—COTANGENTS 


189 





a: 


Tang 
.00000 
.00029 
.00058 
.00087 
.00116 
.00145 
.00175 
00204 
00233 
00262 
00291 
11} .00320 
12; .00349 
13) .00378 
14, .00407 
15) .00436 
16) .00465 
7| .00495 
18} .00524 
19) .00553 
20! .00582 


21; .00611 
22) .00640 
23| .00669 
24} .00698 
25| .00727 
26| .00756 
27; .00785 
28} .00815 
29; .00844 
30| .00873 


31} .00902 
82| .00931 
33| .00960 
34| .00989 
85| .01018 
36) .01047 
37) .01076 
38} .01105 
39| .01135 
40} .01164 


41| .01193 
A42| .01222 
43| .01251 
44) .01280 
45| .01309 
46) .C1338 
47| .01367 
48} .01896 
49; .01425 
50} .01455 


51| .01484 
62; .01513 


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53| .01542 | 


54) 01571 
55| .01600 
56) .01629 
57) .01658 
58| .01687 
59) .01716 
60| .01746 





\ 0° 


Cotang 


Infinite. 


3437.75 
1718.87 
1145.92 
859.436 
687.549 
572.957 
491.106 
429.718 
881.971 
343.774 
812.521 
286.478 
264.441 
245 552 
229.182 
214.858 
202'.219 
190.984 


180.9382 


171.885 


163.700 
156.259 
149.465 
143.237 
137.507 
182.219. 
127.321 
122.77 
118.540 
114.589 


110.892 
107.426 
104.171 
101.107 
98.2179 
95.4895 
92.9085 
90.4633 
88.1436 
85.9398 


83.8435 
81.8470 
79,9434 
78.1263 
76.3900 
74 7292 
73.1390 
71.6151 
0.1533 
68.7501 


67.4019 
66.1055 
64.8580 
63.6567 
62.4992 
61.3829 
60.3058 
59.2659 
58.2612 
57.2900 


1° 


Tang 
.01746 
01775 
.01804 
.01833 
.01862 
.01891 
+-01920 
.01949 
01978 
.02007 
.02036 
.02066 
.02095 
.02124 
.02153 
02182 
02211 
.02240 
02269 
.02298 
02328 


02857 
02386 
02415 
02444 
02473 
02502 
02531 
02560 
102589 
02619 
02648 
02677 
02706 
02735 
02764 
02793 
02822 
02851 
02881 
02910 


.02939 
02968 
02997 
.03026 
03055 
.03084 
03114 
03143 
03172 
03201 


03280 
03259 
03288 
03317 
.03346 
03376 
.03405 
303434 
03463 
03492 





ss Cotang Tang 


89° 


a 





Cotang 











Cotang | 
57.2900 


30.9599 
30.6833 
30.4116 
80.1446 
29.8823. 
29.6245 
29.3711 
29.1220 
28.8771 
28.6363 


88° . 


Tang 




















| 9° 8° 
Tang | Cotang || Tang | Cotang 
.03492 | 28.6363 || .05241 | 19.0811 
.08521 | 28.3994 || .05270 | 18.9755 
.03550 | 28.1664 || .05299 | 18.8711 
.038579 | 27.9372 || .05828 | 18.7678 
.03609 | 27.7117 || .053857 | 18.6656 
.03638 | 27.4899 .05887 | 18.5645 
.03667 | 27.2715 || .05416 | 18.4645 
.03696 | 27.0566 .05445 | 18.8655 
.03725 | 26.8450 .05474 | 18.2677 
.03754 | 26.6367 .05503 | 18.1708 
.03783 | 26.4316 || .05533 | 18.0750 
.03812 | 26.2296 |} .05562 | 17.9802 
.08842 | 26.0807 .05591 | 17.8863 
.03871 | 25.8848 || .05620 | 17.7934 
.03900 | 25.6418 .05649 | 17.7015 
.03929 | 25.4517 .05678 | 17.6106 
.03958 | 25.2644 |) .05708 | 17.5205 
-08987 | 25.0798 .05787 | 17.4314 
.04016 | 24.8978 || .05766 | 17.3432 
.04046 | 24.7185 || .05795 | 17.2558 
.04075 | 24.5418 || .05824 | 17.1693 
.04104 | 24.3675 .05854 | 17.0887 
.04183 | 24.1957 .05883 | 16.9990 
.04162 | 24.0263 || .05912 | 16.9150 
.04191 | 23.8593 || .05941 | 16.8319 
.04220 | 23.6945 .05970 | 16.7496 
.04250 | 23.5821 .05999 ,; 16.6681 
.04279 | 23.3718 .06029 | 16.5874 
.04308 | 23.2187 || .06058 | 16.5075 
.04337 | 23.0577 || .06087 | 16.4288 
.04366 | 22.9038 |] .06116 | 16.3499 
.04395 | 22.7519 || .06145 | 16.2722 
.04424 | 22.6020 || .06175 | 16.1952 
.04454 | 22.4541 .06204 ; 16.1190 
.04483 | 22.3081 .06233 | 16.0435 
.04512 | 22.1640 || .06262 | 15.9687 
.04541 | 22.0217 || .06291 | 15.8945 
.04570 | 21.8813 || .06821 | 15.8211 
.04599 | 21.7426 |} .06350 | 15.7483 
.04628 | 21.6056 || .063879 | 15.6762 
[04658 | 21.4704 || 106408 | 15.6048 
.04687 | 21.3369 || .06437 ) 15.5340 | 
.04716 | 21.2049 -06467 | 15.4638 
.04745 } 21.0747 || .06496 | 15.3943 
.04774 | 20.9460 .06525 | 15.3254 
.04803 | 20.8188 06554 | 15.2571 
-04833 | 20.6932 |} .06584 | 15.1893 
.04862 | 20.5691 .06613 | 15.1222 
.04891 | 20.4465 .06642 | 15.0557 
.04920 | 20.3253 .06671 | 14.9898 
.04949 | 20.2056 || .06700 | 14.9244 
.04978 | 20.0872 || .06730 | 14.8596 
.05007 | 19.9702 .06759 | 14.7954 
.05037 | 19.8546 06788 | 14.7317 
.05066 | 19.7403 || .06817 | 14.6685 
.05095 | 19.6273 .06847 | 14.6059 
.05124 | 19.5156 .06876 | 14.5438 
.05153 | 19.4051 .06905 | 14.4823 
305182 | 19.2959 .06934 | 14.4212 
.05212 | 19.1879 .06963 | 14.3607 
.05241 | 19.0811 .06993 | 14.3007 
|Cotang| Tang ||Cotang| Tang 
87° 

















86° 





= lonwapaasa 








PL ee eee 


| Cotang 


11| .07314 
12| .07344 
13| .07373 
14} .07402 
15| .07431 
16| 07461 
17| .07490 
18} .07519 
19| .07548 
20| .07578 


21} .07607 
22) .07636 
23) .07665 
24| .07695 
25| .07724 
26) .07753 
27| .07782 
28} .07812 
29} .07841 
30| .07870 


31| .07899 








i Cotang 








TABLE XXVIIL—NATURAL TANGENTS 
| ch 


14.3007 
14.2411 
14.1821 
14.1285 
14.0655 
14.0079 
13.9507 
13.8940 
13.8378 
13.7821 
13.7267 


13.6719 
13.6174 
13.5634 
13.5098 
13.4566 
13.4039 
13.3515 
13.2996 
13.2480 
13.1969 


13.1461 
13.0958 
13.0458 
12.9962 
12.9469 
12.8981 
12.8496 
12.8014 
12.7536 
12.7062 


12.6591 
12.6124 
12.5660 
12.5199 
12.4742 
12.4288 
12.3838 
12.3390 
12.2946 
12.2505 


12.2067 
12.1632 


85° 












































Tang 


12278 
123808 
. 12338 
12367 
123897 
12426 
12456 
. 12485 
12515 
12544 
12574 


.12603 
.12633 
12662 
.12692 
12722 
12751 
12781 
.12810 
12840 
12869 


12899 
12929 
12958 
12988 
18017 
1804 

.138076 














| 5° 7 6° 
Tang | Cotang |; Tang | Cotang | 
.08749 | 11.4801 || .10510 | 9.51486 
.08778 | 11.3919 || .10540 | 9.48781 
.08807 | 11.3540 || .10569 | 9.46141 
.08837 | 11.3163 || .10599 | 9.438515 
.O8866 | 11.2789 || .10628.| 9.40904 
.08895 | 11.2417 || .10657 | 9.388307 
.08925 | 11.2048 || .10687 | 9.35724 
.08954 | 11.1681 .10716 | 9.38155 
.08983 | 11.1316 || .10746 | 9.80599 
.09013 | 11.0954 || .10775 | 9.28058 
09042 | 11.0594 || .10805 | 9.25530 
.09071 | 11.0237 || .10834 | 9.23016 
.09101 | 10.9882 || .108638 | 9.20516 
.09130 | 10.9529 || .10898 | 9.18028 
.09159 | 10.9178 |} .10922 | 9.15554 
.09189 | 10.8829 || .10952 | 9.18098 
.09218 | 10.8483 |} .10981 | 9.10646 
.09247 | 10.8189 |} .11011 | 9.08211 
09277 | 10.7797 || .11040 | 9.05789 
.09306 | 10.7457 || .11070 | 9.038379 
.09385 | 10.7119 || .11099 | 9.00988 
.09365 | 10.6788 || .11128 | 8.98598 
.09394 | 10.6450 || .11158 | 8.96227 
09423 | 10.6118 |} .11187 | 8.93867 
,09453 | 10.5789 || .11217 | 8.91520 
.09482 | 10.5462 || .11246 | 8.89185 
.09511 | 10.5186 || .11276 | 8.86862 
.09541 | 10.4813 |} .11805 | 8.84551 
.09570 | 10.4491 .11885 | 8.82252 
.09600 | 10.4172 || .11364 | 8.79964 
.09629 | 10.3854 || .11894 | 8.77689 
.09658 | 10.3538 || .11423 | 8.75425 
.09688 |:10.38224 || .11452 | 8.73172 
09717 | 10.2913 || .11482 | 8.70931 
.09746 | 10.2602 || .11511 | 8.68701 
.09776 | 10.2294 || .11541 | 8.66482 
.09805 | 10.1988 || .11570 | 8.64275 
.09834 | 10.1683 || .11600 | 8.62078 
.09864 | 10.1381 || .11629 | 8.59893 
.09893 | 10.1080 || .11659 | 8.57718 
.09923 | 10.0780 || .11688 | 8.55555 
.09952 | 10.0483 || .11718 | 8.53402 
.09981 | 10.0187 || .11747 | 8.51259 
.10011 | 9.98981 11777 | 8.49128 
.10040 | 9.96007 || .11806 | 8.47007 
.10069 | 9.93101 .11836 | 8.44896 
.10099 | 9.90211 .11865 | 8.42795 
.10128 | 9.87338 || .11895 | 8.40705 
.10158_| 9.84482 || .11924 8.388625 
.10187 | 9.81641 .11954 | 8.36555 
.10216 | 9.78817 || .11983 | 8.34496 
10246 | 9.76009 || .12013 | 8.382446 
10275 | 9.73217 || .12042 | 8.30406 
.10305 | 9.70441 .12072 | 8.283876 
.10334 | 9.67680 || .12101 | 8.26355 
.10363 | 9.64935 || .12181 | 8.24845 
.10393 | 9.62205 || .12160 | 8.22344 
.10422 | 9.59490 || .12190 | 8.20352 
.10452 | 9.56791 12219 | 8.18370 
.10481 | 9.54106 || .12249 | 8.16398 
.10510 | 9.51486 || .12278 | 8.14435 
Cotang| Tang ||Cotang| Tang 
84° 83° 








| 


| 





Cotang 


8.14435 
8.12481 
8.10536 
8.08600 
8.06674 
8.04756 
8.02848 
8.00948 
7.99058 
7.97176 
7.95802 


7.98438 
7.91582 
7.89734 
7.87895 


7.86064 - 


7.84242 
7.82428 
7.80622 
7.78825 
7.77035 


7.75254 
7.73480 
7.71715 
7.69957 
7.68208 
7.66466 
7.64782 
7.63005 
7.61287 
7.59575 


7.57872 
7.56176 
7.54487 
7.52806 
7.51182 
7.49465 
7.47806 
7.46154 
7.44509 
7.42871 


7.41240 
7.39616 
7.87999 
7.86389 
7.34786 
7.33190 
7.81600 
7.30018 
7.28442 
7.26873 


7.25310 


Cotang Tang 
82° 


ae EEEn III EEE IEEE EEEERIIEEIRIREIREREEES ERIE aaa Y 











© 


lommwepnarara 




















AND COTANGENTS 









































; 8° 9° 10° 11° fy 
__|_Tang | Cotang || Tang | Cotang || Tang | Cotang || Tang | Cotang 
, 0} .14054 | 7.11537 || .15838 | 6.31875 || .17633 | 5.67128 || .19485 | 5.14455 |60 
1} .14084 | 7.10038 || .15868 | 6.30189 17663 | 5.66165 || .19468 | 5.18658 |59 
2] .14113 | 7.08546 |; .15898 | 6.29007 7693 | 5.65205 || .19498 | 5.12862 [58 
3| .14143 | 7.07059 |) .15928 | 6.27829 17723 | 5.64248 || .19529 | 5.12069 |57 
4} .14173 | 7.05579 || .15958 | 6.26655 17753 | 5.638295 || .19559 | 5.11279 |56 
5} .14202 | 7.04105 || .15988 | 6.25486 17783 | 5.62544 || .19589 | 5.10490 | 55 
6] .14232 | 7.02637 || .16017 | 6.24821 17813 | 5.61397 || .19619 | 5.09704 |54 
7| .14262 | 7.01174 || .16047 | 6.23160 17843 | 5.60452 || .19649 | 5.08921 |53 
8| .14291 | 6.99718 || .16077 | 6.22003 17873 | 5.59511 || .19680 | 5.08139 |52 
9| .143821 | 6.98268 .16107 | 6.20851 17903 | 5.58573 .19710 | 5.07360 |51 
10| .14351 | 6.96823 || .16137 | 6.19703 17933 | 5.57688 || .19740 | 5.06584 | 50 
11} .14381 | 6.95385 || .16167 | 6.18559 || .17963 | 5.56706 || .19770 | 5.05809 | 49 
12} .14410 | 6.93952 || .16196 | 6.17419 |..17993 | 5.55777 || .19801 | 5.05037 | 48 
13} .14440 | 6.92525 || .16226 | 6.16283 || .18023 | 5.54851 || .198381 | 5.04267 | 47 
14} .14470 | 6.91104 |} .16256 | 6.15151 18053 | 5.53927 || .19861 | 5.03499 | 46 
15} .14499 | 6.89688 || .16286 | 6.14023 18083 | 5.53007 |} .19891 | 5.02734 | 45 
16| .14529 | 6.88278 || .16316 | 6.12899 18113 | 5.52090 || .19921 | 5.01971 | 44 
17| .14559 | 6.86874 || .16346 | 6.11779 || .18143 | 5.51176 || .19952 | 5.01210 |43 
18} .14588 | 6.85475 || .16376 | 6.10664 18173 | 5.50264 |} .19982 | 5.00451 | 42 
19; .14618 | 6.84082 || .16405 | 6.09552 | .18203 | 5.49356 || .20012 | 4.99695 | 41 
20| .14648 | 6.82694 |) .16435 | 6.08444 18233 | 5.48451 |) .20042 | 4.98940 | 40 
21) .14678 | 6.81312 || .16465 | 6.07340 || .18263 | 5.47548 || .20073 | 4.98188 {39 
22| .14707 | 6.79936 || .16495 | 6.06240 | .18293 | 5.46648 || .20103 | 4.97438 | 388 
23| .14737 | 6.78564 || .16525 | 6.05143 || .18323 | 5.45751 || .20133 | 4.96690 | 37 
24| .14767 | 6.77199 || .16555 | 6.04051 || .18353 | 5.44857 || .20164 | 4.95945 | 36 
25| .14796 | 6.75838 || .16585 | 6.02962 |} .183884 | 5.43966 || .20194 | 4.95201 | 35 
26| .14826 | 6.74483 || .16615 | 6.01878 ||} .18414 | 5.48077 || .20224 | 4.94460 | 34 
27| .14856 | 6.73133 || .16645 | 6.00797 || .18444 | 5.42192 || .20254 | 4.93721 133 
28] .14886 | 6.71789 ||} -16674 | 5.99720 || .18474 | 5.41309 |) .20285 | 4.92984 | 382 
29| .14915 | 6.70450 || .16704 | 5.98646 || .18504 | 5.40429 .20315 | 4.92249 | 31 
30} .14945 | 6.69116 || .16734 | 5.97576 18534 | 5.89552 || .20345 | 4.91516 | 30 
31} .14975 | 6.67787 || .16764 | 5.96510 18564 | 5.38677 || .20376 | 4.90785 | 29 
82] .15005 | 6.66463 || .16794 | 5.95448 18594 | 5.37805 || .20406 | 4.90056 | 28 
33} .15034 | 6.65144 || .16824 | 5.94390 18624 | 5.36936 || .20436 | 4.89330 | 27 
34] .15064 | 6.63831 || .16854 | 5.93385 18654 | 5.36070 || .20466 | 4.88605 | 26 
35| .15094 | 6.62523 |} .16884 | 5.92283 18684 | 5.35206 || .20497 | 4.87882 | 25 
36] .15124 | 6.61219 || .16914 | 5.912386 18714 | 5.34345 || .20527 | 4.87162 | 24 
387| .15153 | 6.59921 || .16944 | 5.90191 18745 | 5.83487 || .20557 | 4.86444 | 23 
88) .15183 | 6.58627 || .16974 | 5.89151 18775 | 5.32631 || .20588 | 4.85727 | 22 
39| .15213 | 6.57339 || .17004 | 5.88114 18805 | 5.31778 || .20618 | 4.85013 | 21 
40| .15243 | 6.56055 || .17033 | 5.87080 18835 | 5.30928 || .20648 | 4.84300 | 20 
41| .15272 | 6.54777 |) .17063 | 5.86051 18865 | 5.30080 |} .20679 | 4.83590 | 19 
42| .15302 | 6.535038 ||} .17093 | 5.85024 18895 | 5.29235 || .20709 | 4.82882 | 18 
43| .15332 | 6.52234 || .17123 | 5.84001 18925 | 5.283893 || .20789 | 4.82175 | 17 
44| .15362 | 6.50970 || .17153 | 5.82982 18955 | 5.27553 || .20770 | 4.81471 | 16 
45| .15391 | 6.49710 ||} .17183 | 5.81966 18986 | 5.26715 || .20800 | 4.80769 | 15 
46| .15421 | 6.48456 || .17213 | 5.80953 19016 | 5.25880 |; .208380 | 4.80068 | 14 
47| .15451 | 6.47206 || .17243 | 5.79944 19046 | 5.25048 ||. .20861 | 4.79870 |18 
48| .15481 | 6.45961 || .17273 | 5.78938 19076 | 5.24218 || .20891 | 4.78673 | 12 
49| .15511 | 6.44720 || .17803 | 5.77986 19106 | 5.28391 || .20921 | 4.77978 | 11 
50| .15540 | 6.48484 || .173833 | 5.769387 19136 | 5.22566 |) .20952 | 4.77286 | 10 
51] .15570 | 6.42253 || .173863 | 5.75941 19166 | 5.21744 || .20982 | 4.76595 | 9 
52} .15600 | 6.41026 .17393 | 5.74949 19197 | 5.20925 .21013 | 4.75906 
53; .15630 | 6.39804 || .17423 | 5.73960 19227 | 5.20107 |] .21043-) 4.75219 | 7 
54| .15660 | 6.38587 || .17453 | 5.72974 19257 | 5.19293 || .21073 | 4.74534 | 6 
55| .15689 | 6.87374 || .17488 | 5.71992 || .19287 | 5.18480 || .21104 | 4.738851 | 5 
56| .15719 | 6.36165 || .17513 | 5.71013 19317 | 5.17671 || .21134 | 4.738170 | 4 
57| .15749 | 6.34961 || 9.17543 | 5.70037 19347 | 5.16863 || .21164 | 4.72490 | 3 
58| .15779 | 6.383761 || .17573 | 5.69064 19378 | 5.16058 || .21195 | 4.71813 | 2 
59| .15809 | 6.32566 || .17608 | 5.68094 19408 | 5.15256 || .21225 | 4.771187 | 1 
60} .15838 | 6.81375 || .17633 | 5.67128 19438 | 5.14455 || .21256 | 4.70463 9 
Cotang| Tang | Cotang| Tang ||Cotang| Tang | Cotang| Tang ; 
, 
81° 80° 79° 78° 


a ER AEA SRST NEL ACEC 28 MRI REE ASN PRIETO FOS SIE LEE TEESE OE 


192 TABLE XXVIII.—NATURAL TANGENTS 
12° 13° 14° 15° 


Tang | Cotang || Tang | Cotang || Tang | Cotang|| Tang | Cotang 


0} .21256 | 4.70463 || .23087 | 4.83148 || .24933 | 4.01078 || .26795 | 3.73205 |60 

1| .21286 | 4.69791 || .23117 | 4.32573 || .24964 | 4.00582 || .26826 | 3.72771 |59 

2| .21316 | 4.69121 || .28148 | 4.32001 || .24995 | 4.00086 || .26857 | 3 72338 [58 

3] .213847 | 4.68452 || .23179 | 4.81490 || .25026 | 3.99592 || .26888 | 3.71907 |57 

4| .21377 | 4.67786 || .23209 | 4.30860 || .25056 | 3.99099 |) .26920 | 3.71476 |56 

5] .21408 | 4.67121 || .28240 | 4.80291 || .25087 | 3.98607 || .26951 | 3.71046 |55 

6 

G 

8 

9 
10 








~ 
~ 








21438 | 4.66458 || .23271 | 4.29724 || .25118 | 3.98117 || .26982 | 3.70616 [54 
21469 | 4.65797 || .23801 | 4.29159 || .25149 | 3.97627 || .27013 | 3.70188 |53 
21499 | 4.65138 || .23332 | 4.28595 |) .25180 | 3.97139 || .27044 | 3 69761 |52 
21529 | 4.64480 || .23363 | 4.28032 || .25211 | 3.96651 || .27076 | 3.69335 |51 
21560 | 4.63825 || .238393 | 4.27471 || .25242 | 3.96165 || .27107 | 3.68909 |50 


11} .21590 | 4.63171 || .23424 | 4.26911 || .25273 | 3.95680 || .27188 | 8 68485 |49 
12) .21621 | 4.62518 || .23455 | 4.26352 || .25304 | 3.95196 || .27169 | 3.68061 |48 
13] .21651 | 4.61868 || .23485 | 4.25795 || .25335 | 8.94713 || .27201 | 3.67638 47 | 
14) .21682| 4.61219 || .23516 | 4.25239 || .25366:| 8.94232 || .27282 | 3 67217 46 
15) .21712 | 4.60572 || .23547 | 4.24685 || .25397 | 3.93751 || .27263 | 3.66796 |45 
16} .21743 | 4.5yy27 || .23578 | 4.24182 || .25428 | 3.93271 || .27294 | 3.66376 |44 
17| .21773 | 4.59283 || .23608 | 4.28580 || .25459 | 3.92793 || .27826 | 3.65957 |43 © 
18] .21804 | 4.58641 || .23639 | 4.23080 || .25490 | 3.92316 || .27357 | 3.65538 |42 — 
19| .21834 | 4.58001 || .23670 | 4.22481 || .25521 | 3.91839 || .27388 | 3.65121 |41 
20| .21864 | 4.57363 || .23700 | 4.21933 |) .25552 | 3.91864 |} .27419 | 3.64705 |40 


21} .21895 | 4.56726 || .23731 | 4.21887 || .25583 | 3.90890 || .27451 | 8.64289 |39 
29) .21925 | 4.56091 |) .23762 | 4.20842 || .25614 | 3.90417 || .27482 | 8.63874 |38 © 
23| .21956 | 4.55458 || .23793 | 4.20298 |] .25645 | 3.89945 || .27513 | 3.63461 |37 
24) .21986 | 4.54826 || .23823 | 4.19756 | .25676 | 3.89474 || .27545 | 3.63048 |36 
25| 92017 | 4.54196 || .23854 | 4.19215 |} .25707 | 3.89004 || .27576 | 3.62636 | 35 
96| .22047 | 4.53568 || .23885 | 4.18675 || .25738 | 3.88536 || .27607 | 3.62224 |34— 
27] .22078 | 4.52941 || .23916 | 4.18137 || .25769 | 3.88068 || .27638 | 3.61814 (33 
98] .22108 | 4.52316 || .23946 | 4.17600 || .25800 | 3.87601 |) .27670 | 3.61405 |32 4 
29) .22139 | 4.51698 || .23977 | 4.17064 |} .25831 | 3.87136 || .27701 | 3.60996 (31 © 
30! .22169 | 4.51071 || .24008 | 4.16530 || .25862 | 3.86671 || .27732 | 3 60588 380 


31] .22200 | 4.50451 || .24039 | 4.15997 || .25893 | 3.86208 || .27764 | 3.60181 |29 
32| .222931 | 4.49882 || .24069 | 4.15465 || .25924 | 8.85745 || .27795 | 3.59775 [28 
33] .22261 | 4.49215 || .24100 | 4.14934 || .25955 | 3.85284 || .27826 | 3.59370 |27 
34| .22292 | 4.48600 || .24181 | 4.14405 || .25986 | 3.84824 || .27858 | 3.58966 | 26 
35| .22322 | 4.47986 || .24162 | 4.18877 || .26017 | 3.84364 || .27889 | 3.58562 [25 _ 
36| .22353 | 4.47374 || .24193 | 4.18350 || .26048 | 3.83906 || .27921 | 3.58160 | 24 — 
37| .22383 | 4.46764 || .24223 | 4.12825 26079 | 8.83449 || .27952 | 3.57758 [23 — 
38] .22414 | 4.46155 || .24254 | 4.12301 || .26110 | 3.82992 || .27983 | 3.57357 | 22 — 
39| .22444 | 4.45548 || .24285 | 4.11778 || .26141 | 8.82537 || .28015 | 3.56957 |21 
40| .22475 | 4.44y42 || .24816 | 4.11256 || .26172 | 3.82083 |) .28046 | 3.56557 | 20° 


41} .22505 | 4.44338 || .24347 | 4.10736 |} .26203 | 3.81630 || .28077 | 3.56159 | 19 
42| 22536 | 4.43735 || .24877 | 4.10216 || .26235 | 3.81177 || .28109 | 3.55761 |18 
43| .22567 | 4.43134 || .24408 | 4.09699 || .26266 | 3.80726 || .28140 | 3.55364 [17 
44| ,22597 | 4.42534 || .24439 | 4.09182 || .26297 | 3.80276 || .28172 | 3.54968 |16 
45 22628 | 4.41936 || .24470 | 4.08666 || .26328 | 3.79827 || .28203 | 8.54573 |15 
46 .22658 | 4.41340 || .24501 | 4.08152 || .26859 | 3.'79378 || .28284 | 3.54179 |14 
47 .22689 | 4.40745 |} .24532 | 4.07639 || .26390 | 3.78931 |) .28266 | 3.53785 |13 
48 .22719 | 4.40152 || .24562 | 4.07127 || .26421 78485 || .28297 | 3.53393 | 12 
49, ,22750 | 4.39560 || .24593 | 4.06616 || .26452 78040 || .28329 | 8.53001 |11 
50, .22781 | 4.38969 || .24624 | 4.06107 || .26483 77595 || .28360 | 3.52609 |10 


51) .22811 | 4.38381 || .24655 | 4.05599 || .26515 77152 || .28391 | 3.52219 
52| °.22842 | 4.37793 || .24686 | 4.05092 || .26546 76709 || .28423 | 3.51829 
53| .22872 | 4.87207 || .24717 | 4.04586 |} .2657 76268 || .28454 | 3.51441 
54| .22903 | 4.36623 || .24747 | 4.04081 || .26608 75828 || .28486 | 3.51053 
55| .22934 | 4.36040 || .24778 | 4.08578 |} .26639 75388 || .28517 | 3.50666 
5G6| .22964 | 4.35459 || .24809 | 4.03076 || .26670 74950 || .28549 | 8.50279 
57| .22995 | 4.34879 || .24840 | 4.02574 |] .26701 74512 || .28580 | 3.49894 
58| .23026 | 4.34300 || .24871 | 4.02074 || .26733 | 3.74075 || .28612 | 3.49509 
59| .23056 | 4.83723 || .24902 | 4.01576 || .26764 | 3.73640 || .28643 | 3.49125 
60) .23087 | 4.33148 || .24983 | 4.01078 || .26795 73205 || .28675 | 3.48741 


~ Gotang| Tang | Cotang| Tang | Cotang| Tang | Cotang) Tang j 
74° 


| 77° 16° | 












































C2 G9 G9 G9 CO C9 C9 9 CD OD co 09 Co OO 























—— se 





SHIH TAwWMWHOl 


10 























AND COTANGENTS 












































; 16° 17° 18° 19° 
Tang | Cotang || Tang | Cotang || Tang | Cotang || Tang | Cotang 
.28675 | 3.48741 || .80573 | 8.27085 |) .382492 | 3.07768 || .34433 | 2.90421 | 60 
.28706 | 3.48359 |) .380605 | 3.26745 || .82524 | 3.07464 || .34465 | 2.90147 | 59 
28738 | 3.47977 || .80637 | 3.26406 || .382556 | 3.07160 |/ .84498 | 2.89873 |58 
.28769 | 3.47596 || .380669 | 8.26067 |) .82588 | 3.06857 |} .84530 | 2.89600 | 57 
.28800 | 8.47216 || .80700 | 8.25729 || .82621 | 8.06554 || .384563 | 2.89327 |56 
.28832 | 3.46837 || .30732 | 3.25392 || .82653 | 8.06252 || .34596 | 2.89055 |55 
.28864 | 3.46458 || .80764 | 3.25055 || .382685 | 8.05950 || .34628 | 2.88783 |54 
.28895 | 3.46080 || .380796 | 3.24719 82717 | 3.05649 .384661 | 2.88511 | 53 
.28927 | 8.45703 || .380828 | 3.24383 || .382749 | 3.053849 || .84693 | 2.88240 | 52 
.28958 | 3.453827 || .380860 | 3.24049 || .82782 | 3.05049 || .34726 | 2.87970 |51 
-28990 | 3.44951 || .80891 | 3.23714 || .82814 | 3.04749 || .84758 | 2 87700 |50 
.29021 | 3.44576 || .380923 | 3.23381 |) .82846 | 3.04450 || .384791 | 2.87430 | 49 
-29053 | 3.44202 || .80955 | 3.238048 |, .82878 | 8.04152 |; .84824 | 2.87161 | 48 
.29084 | 3.43829 |} .380987 | 3.22715 || .382911 | 3.03854 || .34856 | 2.86892 | 47 
.29116 | 3.43456 || .81019 | 3.22384 || .82943 | 8.08556 || .84889 | 2.86624 | 46 
.29147 | 3.48084 .81051 | 3.22058 .82975 | 3.038260 || .84922 | 2.86356 | 45 
29179 | 3.42713 || .381088 | 3.21722 || .3838007 | 3.02963 || .84954 | 2.86089 | 44 
~-29210 | 3.42343 || .81115 | 3.21392 || .338040 | 3.02667 || .384987 | 2.85822 | 43 
.29242 | 3.41973 || .31147 | 3.21063 || .33072 | 3.02372 || .85020 | 2.85555 | 42 
.29274 | 3.41604 || .381178 | 3.20784 || .383104 | 3.02077 || .385052 | 2.85289 | 41 
.29305 | 3.41236 || .381210 | 3.20406 || .383136 | 8.01783 || .85085 | 2.85023 | 40 
.29337 | 3.40869 || .31242 | 3.20079 || .33169 | 3.01489 || .385118 | 2.84758 | 39 
.29368 | 3.40502 || .81274 | 3.19752 || .388201 | 3.01196 || .35150 | 2.84494 | 38 
-29400 | 3.40186 || .31306 | 3.19426 || .33233 | 3.00903 || .85183 | 2.84229 | 37 
.29432 | 3.39771 .8138388 | 3.19100 .88266 | 3.00611 .00216 | 2.83896 | 36 
.29463 | 3.39406 || .313870 | 3.1877 .838298 | 3.00819 || .385248 | 2.83702 | 35 
.29495 | 3.39042 || .381402 | 3.18451 || .33380 | 8.00028 || .35281 | 2.83439 | 34 
29526 | 3.38679 || .81484 | 3.18127 || .383363 | 2.99738 || .35314 | 2.88176 | 33 
.29558 | 3.38317 || .31466 | 3.17804 || .3838395 | 2.99447 || .85346 | 2.82914 | 32 
.29590 | 3.37955 || .81498 | 3.17481 || .38427 | 2.99158 || .385379 | 2.82653 | 31 
.29621 | 8.37594 || .31530 | 3.17159 || .88460 | 2.98868 || .35412 | 2.82391 | 30 
.29653 | 3.37234 || .31562 | 3.16838 || .33492 | 2.98580 || .85445 | 2.82130 | 29 
.29685 | 3.36875 .81594 | 3.16517 38524 | 2.98292 85477 | 2.81870 | 28 
29716 | 3.36516 || .31626 | 3.16197 || .388557 | 2.98004 || .35510 | 2.81610 | 27 
“,29748 | 3.36158 || .31658 | 3.15877 || .338589 | 2.97717 35543 | 2.81350 | 26 
-29780 | 3.35800 || .381690 | 3.15558 || .83621 | 2.97480 || .85576 | 2.81091 | 25 
29811 | 3.35448 || .381722 | 3.15240 || .33654 | 2.97144 || .35608 | 2.80833 | 24 
.29843 | 8.35087 || .31754 | 3.14922 || .33686 | 2.96858 || .35641 | 2.80574 | 23 
.29875 | 3.34732 || .81786 | 3.14605 || .83718 | 2.96573 || «85674 | 2.80316 | 22 
.29906 | 3.343877 || .381818 | 3.14288 || .338751 | 2.96288 || .385707 | 2.80059 | 21 
.29938 | 8.34023 || .81850 | 3.18972 || .83783 | 2.96004 || .85740 | 2.79802 | 20 
-29970 | 3.33670 || .81882 | 3.18656 |) .383816 | 2.95721 |) .85772 | 2.79545 | 19 
.30001 | 3.33317 || .31914 | 3.13841 || .388848 | 2.95437 || .85805 | 2.79289 | 18 
.80033 | 3.32965 || .81946 | 3.13027 |} .83881 | 2.95155 || .85838 | 2.79083 |17 
.80065 | 3.32614 || .81978 | 3.12718 || .383918 | 2.94872 || .385871 | 2.78778 | 16 
.80097 | 3.32264 .82010 | 8.12400 83945 | 2.94591 .85904 | 2.785238 | 15 
380128 | 3.381914 |} .82042 | 3.12087 || .83978 | 2.94809 || .85937 | 2.78269 | 14 
.80160 | 3.31565 || .382074 | 3.1177 84010 | 2.94028 || .385969 | 2.78014 |13 
.30192 | 3.31216 || .82106 | 3.11464 || .34043 | 2.93748 || .36002 | 2.77761 | 12 
. 30224 | 3.30868 || .82139 | 3.11153 || .34075 | 2.93468 || .36085 | 2.77507 | 11 
.80255 | 3.30521 || .82171 | 3.10842 || .384108 | 2.93189 |} .86068 | 2.77254 | 10 
80287 | 3.30174 || .82203 | 3.10532 || .84140 | 2.92910 || .36101 | 2.77002 | 9 
.80319 | 3.29829 || .82285 | 3.10223 || .34173 | 2.92632 || .36134 | 2.76750 | 8 
80351 | 3.29483 || .82267 | 3.09914 |} .384205 | 2.92354 || .36167 | 2.76498 | 7 
.30382 | 3.29139 || .82299 | 3 09606 || .34238 | 2.92076 || .86199 | 2 76247 | 6 
.80414 | 3.28795 82331 | 3.09298 || .84270 | 2.91799 || .862382 | 2 75996 | 5 
80446 | 3.28452 || .82363 | 3.08991 |} .34303 | 2.91523 || .386265 | 2 75746 | 4 
30478 | 3.28109 || .82396 | 3.08685 |} 84335 | 2.91246 || .36298 | 2.75496 | 3 
.80509 | 8.27767 .82428 | 8.08879 || .3843868 | 2.90971 .86331 | 2.75246 | 2 
80541 | 8.27426 || .32460 | 8.08073 || .34400 | 2.90696 || .36364 | 2.74997 | 1 
.80573 | 3.27085 || .82492 | 3.07768 || .34433 | 2.90421 || .36397 | 2.74748 | 0 
Cotang| Tang ||Cotang| Tang ||Cotang| Tang ||Cotang/ .Tang ; 

73° 72° (he 70° 





a 


194 TABLE XXVIII.—NATURAL TANGENTS 





| 


22° 


Tang | Cotang 





























20° | Q1° 
Tang | Cotang |} Tang | Cotang 
0| .36397 | 2.74748 || .38386 | 2.60509 
1) .86480 | 2.74499 || .388420 | 2.60283 
2} .36463 | 2.74251 || .388453 | 2.60057 
3; .86496 | 2.74004 || .88487 | 2.59831 
4 .86529 | 2.78756 || .388520 | 2.59606 
5) .86562 | 2.73509 || .388553 | 2.593881 
6| .86595 | 2.73268 ||} .88587 | 2.59156 
7| .86628 | 2.73017 || .88620 | 2.58932 
8 .36661 | 2.72771 || .88654 | 2.58708 
9, .86694 | 2.72526 || .388687 | 2.58484 
10| .86727 | 2.72281 || .88721 | 2.58261 
11| .36760 | 2.72036 || .388754 | 2.58038 
12) .86793 | 2.71792 || .88787 | 2.57815 
13) .86826 | 2.71548 || .88522 | 2.57593 
14) .36859 | 2.71805 || .88854 | 2.57371 
15) .36892 | 2.71062 || .88888 | 2.57150 
16} .36925 | 2.70819 || ,388921 | 2.56928 
17} .36958 | 2.7057 .88955 | 2.56707 
18| .36991 | 2.70335 || .88988 | 2.56487 
19} .387024 | 2.70094 || .39022 | 2.56266 
20, .87057 | 2.69858 || .89055 | 2.56046 
21] .87090 | 2.69612 || .389089 |) 2.55827 
22) 371238 | 2.69371 || ,389122 | 2.55608 
23| .87157 | 2.69131 || .39156 | 2.55389 
24| .87190 | 2.68892 || .89190 | 2.55170 
25| .37223 | 2.68653 || ,389223 | 2.54952 
26) .87256 | 2.68414 || .389257 | 2.54734 
27| .87289 | 2.68175 || .389290 | 2.54516 
28] .373822 | 2 67987 || .3893824 | 2.54299 
29| .387355 | 2.67700 || .89357 | 2.54082 
30| .37888 | 2.67462 || 389891 | 2.53865 
31| .87422 | 2.67225 || .389425 | 2.53648 
82] .87455 | 2.66989 || .39458 | 2.53432 
83] .87488 | 2.66752 || .389492 | 2.53217 
34| .87521 | 2.66516 || .89526 | 2.53001 
35| .37554 | 2.66281 || .389559 | 2.52786 
36| .37588 | 2.66046 || .39593-| 2.5257 
37| .37621 | 2.65811 || .389626 | 2.52357 
38| .87654 | 2.65576 || .89660 | 2.52142 
39| .37687 | 2.65342 || .89694 | 2.51929 |, 
40| .87720 | 2.65109 || .89727 | 2.51715 
41) .37'754 | 2.64875 || .89761 | 2.51502 
42| 37787 | 2.64642 || .39795 | 2.51289 
43} 37820 | 2.64410 || .39829 | 2.51076 
44| .87853 | 2.64177 || .89862 | 2.50864 
45) .37887 | 2.63945 || .39896 | 2.50652 
46| .37920 | 2.63714 || .39980 | 2.50440 
47| .37953 | 2.63483 || .389963 | 2.50229 
48] .37986 | 2.63252 || .89997 | 2.50018 
49|- .88020 | 2.63021 || .40031 | 2.49807 
50| .38053 | 2.62791 || .40065 | 2.49597 
51) .38086 | 2.62561 |} .40098 | 2.49386 
52] .38120 | 2.62332 || .40182 | 2.49177 
53| .38153 | 2.62108 || .40166 | 2.48967 
54] .88186 | 2.61874 || .40200 | 2.48758 
55| .38220 | 2.61646 || .40234.| 2.48549 
56| .38253 | 2.61418 || .40267 | 2 48340 
57| .88286 | 2.61190 || .40801 | 2.48132 
58] .388320 | 2.60963 || .40385 | 2.47924 
59| .88353 | 2.60736 || .40369 | 2.47716 
60| .38386 | 2.60509 || .40403 | 2.47509 
Cotang| Tang ||Cotang| Tang 
1 Vel) | peace ee at 
69° 68° 


a Tae a ie Na a eer 


























.40403 | 2.47509 
.40486 | 2.47302 
.40470 | 2.47095 
.40504 | 2.46888 
.40588 | 2.46682 
.40572 | 2.46476 
-40606 | 2.46270 
-40640 | 2.46065 
.40674 | 2.45860 
40707 | 2.45655 
40741 | 2.45451 


40775 | 2.45246 
-40809 | 2.45043 
.40843 | 2.44839 
-40877 | 2.44636 
40911 | 2.44438 
40945 | 2.44230 
.40979 | 2.44027 
.41013 | 2.48825 
-41047 | 2.48623 
.41081 | 2.48422 


41115 | 2.48220 
.41149 | 2.43019 
41183 | 2.42819 
41217 | 2.42618 
41251 | 2.42418 
41285 | 2.42218 
.41819 | 2.42019 
.41353 | 2.41819 
.41387 | 2.41620 
41421 | 2.41421 


.41455 | 2.41223 
.41490 | 2.41025 
41524 | 2.40827 
.41558 | 2.40629 
.41592 | 2.404382 
.41626 | 2.402385 
.41660 | 2.40088 
.41694 | 2.39841 
41728 | 2.89645 
.41763 | 2.39449 


41797 | 2.39258 
.41831 | 2.89058 
.41865 | 2.388863 

41899 | 2.38668 
.41933 | 2.38473 
.41968 | 2.38279 
.42002 | 2.388084 
-42036 | 2.37891 
.42070 | 2.37697 
.42105 | 2.37504 


421389 | 2.37311 
42173 | 2.87118 
42207 | 2.86925 
42242 | 2.36733 
42276 | 2.36541 
.42310 | 2.86349 
42345 | 2.36158 
42379 | 2.385967 
42413 | 2.35776 
42447 | 2.35585 





Cotang| Tang 


67° 


| 














23° 
Tang | Cotang 


42447 | 2.85585 | 60 
42482 | 2.35395 |59 
-42516 | 2.85205 |58 
-42551 | 2.85015 |57 
.42585 | 2.384825 |56 
-42619 | 2.384686 | 55 
-42654 | 2.34447 |54 
-42688 | 2.384258 |53 
42722 | 2.34069 | 52 
42757 | 2.33881 | 51 
42791 | 2.33693 |50 


-42826 | 2.33505 |49 
-42860 | 2.83317 |48 
42894 | 2.33130 |47 
42929 | 2.382943 |46 
.42963 | 2.382756 |45 
-42998 | 2.82570 | 44 
43032 | 2.82383 |438 
.43067 | 2.82197 |42 
.43101 | 2.82012 |41 
-43186 | 2.31826 |40 — 


43170 | 2.81641 |39 
-43205 | 2.31456 |38 
43239 | 2.31271 137 
43274 | 2.31086 |36 
-43308 | 2.80902 |85 
43343 | 2.30718 |84 
43378 | 2.30534 |83 
43412 | 2.30351 [82 
43447 | 2.30167 |81 - 
43481 | 2.29084 |30 


43516 | 2.29801 |29 _ 
"43550 | 2.29619 |28 
"43585 | 2.29487 197 
"43620 | 2.29954 |26 
"43654 | 2.29073 |25 
"43689 | 2.28891 |24 
"43724 | 2.28710 |23- 
"43758 | 2.28528 |22- 
"43798 | 2.28348 [21 
"43928 | 2.28167 | 20 


43862 | 2.27987 |19 
43897 | 2.27806 |18 
43932 | 2.27626 17 
.43966 | 2.27447 |16 
.44001 | 2.27267 | 15 
.44036 | 2.27088 |14 
.44071 | 2.26909 |13— 
.44105 | 2.26730 |12 
.44140 | 2.26552 |11 
44175 | 2.26874 | 10 


.44210 | 2.26196 | 9 
.44244 | 2,26018 
.44279 | 2.25840 
.44314 | 2.25663 
.443849 | 2.25486 
44384 | 2.25309 
.44418 | 2.25182 
.44453 | 2.24956 
.44488 | 2.24780 
44593 | 2.24004 


Cotang| Tang 
66° 














int | OM COR OT 300 








AND COTANGENT'S 














: 24° | 25° 26° 
Tang | Cotang |; Tang | Cotang || Tang | Cotang 
0} .44523 | 2.24604 || .46631 | 2.14451 || .48773 | 2.05030 
1) .44558 | 2.24428 || .46666 | 2.14288 || .48809 | 2.04879 
2| .44593 | 2.24252 || .46702 | 2.14125 || .48845 | 2.04728 
8} .44627 | 2.24077 || .46737 | 2.13963 || .48881 | 2.04577 
4| .44662 | 2.28902 || .46772 | 2.13801 || .48917 | 2.04426 
5| .44697 | 2.23727 || .46808 | 2.13639 || .48953 | 2.04276 
6| .447382 | 2.23553 || .46843 | 2.13477 || .48989 | 2.04125 
7| .44767 | 2.23878 || .46879 | 2.13316 || .49026 | 2.03975 
8} .44802 | 2.23204 || .46914 | 2.13154 || .49062 | 2.03825 
9| .44837 | 2.23030 || .46950 | 2.12993 || .49098 | 2.03675 
10) .44872 | 2.22857 || .46985 | 2.12882 || .49184 | 2.03526 
11| .44907 | 2.22683 |) .47021 | 2.12671 || .49170 | 2.03376 
12} .44942 | 2.22510 || .47056 | 2.12511 || .49206 | 2.03227 
13} .44977 | 2.22337 || .47092 | 2.12350 || .49242 | 2.03078 
14) .45012 | 2.22164 || .47128 | 2.12190 |} .49278 | 2.02929 
15| .45047 | 2.21992 || .47163 | 2.12030 || .493815 | 2.02780 
16} .45082 | 2.21819 || .47199 | 2.11871 || .49851 | 2.02631 
17| .45117 | 2.21647 || .47284 4 2.11711 || .49387 | 2.02483 
18) .45152 | 2.21475 || .47270 | 2.11552 || .49423 | 2.02335 
19} .45187 | 2.21304 || .47805 | 2.11892 || .49459 | 2.02187 
20} .45222 | 2.21182 || .473841 | 2.11233 |} .49495 | 2.02039 
21) .45257 | 2.20961 || .47377 | 2.11075 || .495382 | 2.01891 
22| .45292 | 2.20790 || .47412 | 2.10916 || .49568 | 2.01743 
23| .453827 | 2.20619 || .47448 | 2.10758 || .49604 | 2.01596 
24| .45362 | 2.20449 || .47483 | 2.10600 || .49640 | 2.01449 
25| 45397 | 2.20278 || .47519 | 2.10442 |) .49677 | 2:01302 
26| .45482 | 2.20108 |} .47555 | 2.10284 || .49713 | 2.01155 
27| .45467 | 2.19938 || .47590 | 2.10126 || .49749 | 2.01008 
28| .45502 | 2.19769 || .47626 | 2.09969 || .49786 | 2.00862 
29| .455388 | 2.19599 || .47662 | 2.09811 || .49822 | 2.00715 
30} .45573 | 2.19480 |} .47698 | 2.09654 || .49858 | 2.00569 
81} .45608 | 2.19261 || .47733 | 2.09498 |} .49894 | 2.00423 
32| .45643 | 2.19092 || .47769 | 2.09341 || .49931 | 2.00277 
83} .45678 | 2.18923 || .47805 | 2.09184 || .49967 | 2.00131 
34 .45713 | 2.18755 || -47840 | 2.09028 || .50004 | 1.99986 . 
385; .45748 | 2.18587 || .47876 | 2.08872 || .50040 | 1.99841 
36] .45784 | 2.18419 || .47912 | 2.08716 || .50076 | 1.99695 
37) .45819 | 2.18251 || .47948 | 2.08560 || .50113 | 1.99550 
38) .45854 | 2.18084 |} .47984 | 2.08405 || .50149 | 1.99406 
89| .45889 | 2.17916 |} .48019 | 2.08250 || .50185 | 1.99261 
40| .45924 | 2.17749 || .48055 | 2.08094 || .50222 | 1.99116 
41| .45960 | 2.17582 |} .48091 | 2.07939 || .50258 | 1.98972 
42, .45995 | 2.17416 || .48127 | 2.07785 || .50295 | 1.98828 
43| .46030 | 2.17249 || .48163 | 2.07680 || .50381 | 1.98684 
44| .46065 | 2.17083 || .48198 | 2.07476 || .50368 | 1.98540 
45; .46101 | 2.16917 || .48234 | 2.07321 || .50404 | 1.98396 
46 .46186 | 2.16751 || .48270 | 2.07167 || .50441 | 1.98253 
47| .46171 | 2.16585 || .48806 | 2.07014 || .50477 | 1.98110 
48 .46206 | 2.16420 || .48342 , 2.06860 || .50514 | 1.97966 
49 .46242 | 2.16255 || .48878 | 2.06706 || .50550 |.1.97823 
50, .46277 | 2.16090 || .48414 | 2.06553 || .50587 | 1.97681 
51. .46312 | 2.15925. |) .48450 | 2.06400 || .50623 | 1.97538 
52| .46348 | 2.15760 || .48486 | 2.06247 || .50660 | 1.97395 
53} .46383 | 2.15596 || .48521 | 2.06094 || .50696 | 1.97253 
54! .46418 | 2.15432 || .48557 | 2.05942 || .50783 | 1.97111 
55| .46454 | 2.15268 || .48593 | 2.05790 |) .50769 | 1.96969 
56; .46489 | 2.15104 || .48629 | 2.05637 || .50806 | 1.96827 
57) .46525 | 2.14940 || .48665 | 2.05485 || .50843 | 1.96685 
58! .46560 | 2.14777 || 748701 | 2.05333 || .50879 | 1.96544 
59| .46595 | 2.14614 || .48737 | 2.05182 |; .50916 | 1.96402 
60| .46631 | 2.14451 || .48773 | 2.05030 |) .50953 | 1.96261. 
Cotang| Tang ||Cotang, Tang |,Cotang| Tang 
, —— Ee eee 
65° 64° 63° 










































































27° : 
Tang | Cotang 
50953 | 1.96261 | 60 
.50989 | 1.96120 |59 
.51026 | 1.95979 58 
.51063 | 1.95888 |57 
.51099 | 1.95698 56 
51186 | 1.95557 | 55 
651173 | 1.95417 |54 
-51209 | 1.95277 | 53 
.51246 | 1.95137 |52_ 
-51283 | 1.94997 |51 
51319 1.94858 |50 
.51356 | 1.94718 |49 
-51393 | 1.94579 | 48 
.51430 | 1.94440 | 47 
.51467 | 1.94301 | 46 
.51503 | 1.94162 45 
.51540 | 1.94023 44 
-51577 | 1.93885 43 
.51614 | 1.93746 42 
51651 | 1.93608 41 
.51688 | 1.93470 | 40 
.91724 | 1.93332 |39 
51761 | 1.93195 | 38 
.51798 | 1.93057 | 37 
.51835 | 1.92920 |36 
51872 | 1.92782 | 35 
-51909 | 1.92645 | 34 
-51946 | 1.92508 |33 
.51983 | 1.92871 |32 
.52020 | 1.92285 |31 
52057 | 1.92098 | 30 
52094 | 1.91962 | 29 
_ 52131 | 1.91826 (28 
-52168 | 1.91690 | 27 
.52205 | 1.91554 | 26 
.52242 | 1.91418 |25 
52279 | 1.91282 | 24 
52316 | 1.91147 |23 
.52353 | 1.91012 |22 
.52390 | 1.90876. 21 
52427 | 1.90741 20 
52464 | 1.90607 19 
52501 | 1.90472 | 18 
52588 | 1.90387 | 17 
.52575 | 1.90203 |16 
.52613 | 1.90069 |.15 
.52650 | 1.89935 | 14 
52687 | 1.89801 (18 
52724 | 1.89667 112 
52761 | 1.89533 | 11 
52798 | 1.89400 | 10 
.52836 | 1.89266 | 9 
52873 | 1.89188 | 8 
.52910 | 1.89000 | 7 
52947 | 1.88867 | 6 
52985 | 1.88734 | 5 
53022 | 1.88602 | 4 
53059 | 1.88469 | 3 
53096 | 1.883387 | 2 
53134 | 1.88205 | 1 
53171 | 1.88073 | 0 
Cotang| Tang i 
62° 


196 TABLE XXVIII.—NATURAL TANGENTS 



























































i 28° | 29° 30° ae ; 
Tang | Cotang || Tang | Cotang | Tang Cotang || Tang | Cotang 
0} .538171 | 1.88073 .00431 | 1.80405 || .57785 | 1.78205 || .60086 | 1.66428 |60 
1| .538208 | 1.87941 .55469 | 1.80281 || .57774 | 1.73089 || .60126 | 1.66318 |59 
2| .53246 | 1.87809 || .55507 | 1.80158 || .57813 | 1.72973 || .60165 | 1.66209 158 
3] .58283 | 1.87677 || .55545 | 1.80084 || .57851 | 1.72857 || .60205 | 1.66099 |57 
4! .53320 | 1.87546 || .55588 | 1.79911 || .57890 | 1.72741 .60245 | 1.65990 |56 
5| .58858 | 1.87415 || .55621 | 1.79788 || .57929 | 1.72625 || .60284 | 1.65881 155 
6| .58395 | 1.87283 || .55659 | 1.79665 || .57968 | 1.72509 || .60824 | 1.65772 | 54 
7| .58482 | 1.87152 || .55697 | 1.79542 || .58007 | 1.72893 || .60864 | 1.65663 [53 
8) .538470 | 1.87021 ||} .55736 | 1.79419 || .58046 | 1.72278 || .60403 | 1.65554 |52 
~ 9} .58507 | 1.86891 || .55774 | 1.79296 || .58085 | 1.72163 || .60448 | 1.65445 151 
10} .538545 | 1.86760 || .55812 | 1.79174 || .58124 | 1.72047 || .60483 | 1.65387 | 50 
11] .53582 | 1.86630 || .55850 | 1.79051 .58162 | 1.71932 || .60522 | 1.65228 | 49 
12) .53620 | 1.86499 || .55888 | 1.78929 || .58201 | 1.71817 || .60562 | 1.65120 | 48 
18) .58657 | 1.86369 .55926 | 1.78807 .58240 | 1.71702 || .60602 | 1.65011 | 47 
14] .53694 | 1.86289 || .55964 | 1.78685 || .58279 | 1.71588 || .60642 | 1.64903 |46 
15;| .58732 | 1.86109 .56008 | 1.78563 .58318 | 1.71473 .60681 | 1.64795 | 45 
16) .53769 | 1.85979 |; .56041 | 1.78441 .58357 | 1.71858 || .60721 | 1.64687 | 44 
17| .58807 | 1.85850 || .56079 | 1.78319 || .58896 |. 1.71244 || .60761 | 1.64579 | 438 
18) -.58844 | 1.85720 .56117 | 1.78198 .58435 | 1.71129 .60801 | 1.64471 | 42 
19; .538882 | 1.85591 .56156 | 1.78077 || .5847 1.71015 || .60841 | 1.64863 |41 
20; .53920 | 1.85462 || .56194 | 1.77955 || .58513 | 1.70901 || .60881 | 1.64256 | 40 
21| .58957 | 1.853383 || .56282 | 1.77884 || .58552 | 1.70787 || .60921 | 1.64148 |39 
22] .58995 | 1.85204 || .56270 | 1.77713 || .58591 | 1.70678 || .60960 | 1.64041 |38 
23) .54082 | 1.85075 || .56809 | 1.77592 || .58631 | 1.70560 || .61000 | 1.63934 | 37 
24| .54070 | 1.84946 || .56347 | 1.77471 .58670 | 1.70446 || .61040 | 1.63826 |36 
25| .54107 | 1.84818 || .56385 | 1.77351 .58709. | 1.70332 || .61080 | 1.63719 | 35 
26| .54145 | 1.84689 || .56424 | 1.77280 || .58748 | 1.70219 || .61120 | 1.63612 |34 
27) .54183 | 1.84561 || .56462 | 1.77110 || .58787 | 1.70106 || .61160 | 1.63505 | 33 
28) .54220 | 1.84483 || .56501 | 1.76990 || .58826 | 1.69992 || .61200 | 1.63398 |32 
29! 54258 | 1.84805 .56539 | 1.76869 .58865 | 1.69879 .61240 | 1.63292 |31 
30) -.54296 | 1.84177 || .56577 | 1.76749 || .58905 | 1.69766 || .61280 | 1.63185 | 30 
81| .54883 | 1.84049 || .56616 | 1.76629 || .58944 | 1.69653 || .61320 | 1.63079 | 29 
82| .54371 | 1.83922 || .56654 | 1.76510 || .58983 | 1.69541 .613860 | 1.62972 | 28 
83/ .54409 | 1.88794 |} .566938 | 1.76390 || .59022 | 1.69428 || .61400 | 1.62866 |27 
84| 154446 | 1.83667 || .56731 | 1.76271 || .59061 | 1.69316 || .61440 | 1.62760 |26 — 
85| .54484 | 1.83540 || .56769 1.76151 | .69101 | 1.69203 || .61480 | 1.62654 | 25 
86] .54522 | 1.838413 || .56808 | 1.76082 || .59140 | 1.69091 || .61520 | 1.62548 |24 
87| .54560 | 1.88286 || .56846 1.75913 .59179 | 1.68979 || .61561 | 1.62442°|23 © 
88; .54597 | 1.83159 |) .56885 | 1.75794 || .59218 | 1.68866 |} .61601 | 1.62386 | 22 © 
89) .54685 | 1.83033 || .56923 | 1.75675 || .59258 | 1.68754 || .61641 | 1.62230 | 21 . 
40| .54673 | 1.82906 || .56962 | 1.75556 | .59297 | 1.68643 ||} .61681 | 1.62125 | 20 — 
41) .54711 | 1.82780 || .57000 | 1.75487 || .59336 | 1.68531 || .61721 | 1.62019 |19 © 
42| .54748 | 1.82654 || .57089 | 1.75319 || .59876 | 1.68419 || .61761 | 1.61914 |18 
43) .54786 | 1.82528 || .57078 | 1.75200 || .59415 | 1.68808 || .61801 | 1.61808 |17 
44| .54824 | 1.82402 || .57116 | 1.75082 || .59454 | 1.68196 || .61842 | 1.61703 |16 : 
45| .54862 | 1.82276 || .57155 | 1.74964 || .59494 | 1.68085 |} .61882 | 1.61598 |15 © 
46| ©4900 | 1.82150 || .57193 | 1.74846 || .59533 | 1.67974 || .61922 | 1.61493 |14 
47| .54938 | 1.82025 || .57282 | 1.74728 || .59573 | 1.67863 || .61962 | 1.61388 |13 q 
48| .54975 | 1.81899 || .57271 | 1.74610 ||} .59612 | 1.67752 || .62003 | 1.61283 112 
49| .55013 | 1.81774 || .573809 | 1.74492 || .59651 | 1.67641 || .62043 | 1.61179 |11 
50| .55051 | 1.81649 || .57848°) 1.74375 || .59691°| 1.67520 || .62083 | 1.61074 |10- 
51] .55089 | 1.81524 || .57886 | 1.74257 |) .59730 | 1.67419 || .62124 | 1.60970 | 9 
52) .55127 | 1.81399 || .57425 | 1.74140 || .59770 | 1.67309 || .62164 | 1.60865 | 8 
53| .55165 | 1.81274 || .57464 | 1.74022 || .59809 | 1.67198 || .62204 | 1.60761 | 7 
54}. .55203 | 1.81150 || .575038 | 1.78905 || .59849 | 1.67088 || .62245 | 1.60657 | 6 
55| .55241 | 1.81025 .57541 | 1.73788 .59888 | 1.66978 .62285 | 1.60553 | 5 
56) .55279 | 1-.80901 .57580 | 1.73671 || .59928 | 1.66867 || .62325 | 1.60449 | 4 
“| .553817 | 1.8077’ .57619 | 1.738555 || .59967 | 1.66757 || .62366 | 1.60345 | 3 
58} .55355 | 1.80653 || .57657 | 1.73488 || .60007 | 1.66647 || .62406 | 1.60241.| 2 
59] .55393 | 1.80529 || .57696 | 1.73321 |) .60046 | 1.66538 || .62446 | 1.60137 | 1 
60| .55431 | 1.80405 Dir taoe | LevecOd .60086 | 1.66428 .62487 | 1.600383 | 0 
¢ Cotang| Tang ||Cotang| Tang |,Cotang| Tang |/Cotang| Tang a 
61° 60° | 59° 58° 

















S et 
TSOOIDONP WOH OS 











32° 
_Tang_ Cotang 
2487 | 1.60033 
"62527 | 1.59930 
.62568 | 1.59826 
.62608 | 1.59728 
.62649 | 1.59620 
-62689 | 1.59517 
.62730 | 1.59414 
.62770 | 1.59811 
.62811 | 1.59208 
.62852 | 1.59105 
-62892 | 1.59002 
.629383 | 1.58900 
.62973 | 1.58797 
.63014 | 1.58695 
.63055 | 1.58593 
.68095 | 1.58490 
.63186 | 1.58388 
.63177 | 1.58286 
.63217 | 1.58184 
. 63258 perce 
.63299 | 1.57981 
.63340 | 1.57879 | 
.63380 | 1.5777 
.63421 | 1.57676 
.63462 | 1.57575 
.635038 | 1.57474 
.63544 | 1.57872. 
.63584 | 1.57271 
. 63625 1 Bt¥O 
.63666 | 1.57069 
.638707 | 1.56969 
.63748 | 1.56868 
.63789 | 1.56767 
.638380 | 1.56667 
.63871 | 1.56566 
.63912 | 1.56466 
.63953 | 1.56366 
.638994 | 1.56265 
.64035 | 1.56165 
.64076 | 1.56065 
-64117 | 1.55966 
.64158 | 1.55866 
.64199 | 1.55766 
.64240 | 1.55666 
.64281 | 1.55567 
.64322 | 1.55467 
.64863 | 1.55368 
.64404 | 1.55269 
.64446 | 1.55170 
.64487 | 1.55071 
.64528 | 1.54972 
.64569 | 1.54873 
.64610 | 1.54774 
.64652 | 1.54675 
.64693 | 1.54576 
.64734 | 1.54478 
64775 | 1.54379. 
.64817 | 1.54281 
-64858 | 1.54183 
.64899 | 1.54085 
.64941 | 1.53986 


- |Cotang| Tang 





57° 




















- AND COTANGENTS 
























































33° 34° | 35° 

Tang | Cotang || Tang | Cotang | Tang | Cotang 
.64941 | 1.53986 |} .67451 | 1.48256 || .70021 | 1.42815 
-64982 | 1.53888 || .67493 | 1.48163 | .70064 | 1.42726 
65024 | 1.53791 67536 | 1.48070 || .70107 | 1.42688 
.65065 | 1.53693 67578 | 1.47977 || .70151 | 1.42550 
.65106 | 1.53595 67620 | 1.47885 || .70194 | 1.42462 
.65148 | 1.53497 67663 | 1.47792 || .70288 | 1.42374 
.65189 | 1.53400 67705 | 1 "47699 70281 | 1.42286 
.65231 | 1.53302 67748 | 1.47607 |) .70325 | 1.42198 
.65272 | 1.58205 67790 | 1.47514 || .70868 | 1.42110 
.65314 | 1.53107 67832 | 1.47422 || .70412 | 1.42022 
.65355 | 1.53010 67875 | 1.47330 ||. .70455 | 1.41934 
65397 | 1.52913 67917 | 1.47288 || .70499 | 1.41847 
.65488 | 1.52816 | .67960 | 1.47146 || .70542 | 1.41759 
.65480 | 1.52719 68002 | 1.47053 || .70586 | 1.41672 
.65521 | 1.52622 68045 | 1.46962 || .70629 | 1.41584 
.65563 | 1.52525 68088 | 1.46870 || .'70673 | 1.41497 
.65604 | 1.52429 || .68180 | 1.4677 -VOT1T | 1.41409 
.65646 | 1.52332 68173 | 1.46686 || .70760 | 1.41322 
.65688 | 1.52285 || .68215 | 1.46595 || .70804 | 1.41235 
-65729 | 1.52139 68258 | 1.46503 || .70848 | 1.41148 
65771 | 1.52048 68301 | 1.46411 || .70891 | 1.41061 
.65813 | 1.51946 68343 | 1.46320 || .70935 | 1.40974 
.65854 | 1.51850 68386 | 1.46229 || .70979 | 1.40887 
.65896 | 1.51754 68429 | 1.46137 || .71023 | 1.40800 
.65988 | 1.51658 68471 | 1.46046 || .'71066 | 1.40714 
.65980 | 1.51562 68514 | 1.45955 || .71110 | 1.40627 
66021 |} 1.51466 68557 | 1.45864 || .71154 | 1.40540 
.66063 | 1.51370 68600 | 1.45773 || .'71198 | 1.40454, 
66105 | 1.51275 68642 | 1.45682 || .71242 | 1.40367 
66147 | 1.51179 68685 | 1.45592 |) .71285 | 1.40281 
.66189 | 1.51084 |] .68728 | 1.45501 || .713829 | 1.40195 
66280 | 1.50988 68771 | 1.45410 || .71873 | 1.40109 
.66272 | 1.50898 68814 | 1.45320 || .71417 | 1.40022 
.66314 | 1.50797 68857 | 1.45229 || .71461 | 1.39936 
- 66856 | 1.50702 68900 | 1.45139 || .71505 | 1.39850 
.66398 | 1.50607 68942 | 1.45049 || .71549 | 1.39764 
.66440 | 1.50512 68985 | 1.44958 || .71593 | 1.39679 
.66482 | 1.50417 || .69028 | 1.44868 || .71637 }| 1.39593 
66524 | 1.50822 || .69071 | 1.44778 || .71681 | 1.39507 
.66566. | 1.50228 69114 | 1.44688 || .71725 | 1.39421 
.66608 | 1.50133 || .69157 | 1.44598 || .71'769 | 1.39336 
.66650 | 1.50038 || .69200 | 1.44508 || .71813 | 1.389250 
.66692 | 1.49944 || .69243 | 1.44418 |) .71857 | 1.39165 
.66734 | 1.49849 || 69286 | 1.44829 || .71901 | 1.39079 
66776 | 1.49755 |) .69329 | 1.44239 || .71946 | 1.38994 
.66818 | 1.49661 || .69372 | 1.44149 || .71990 | 1.38909 
.66860 | 1.49566 || .69416 | 1.44060 || .72084 | 1.38824 
.66902 | 1.49472 || .69459 | 1.43970 || .72078 | 1.38738 
.66944 | 1.49378 || .69502 | 1.43881 12122 | 1.38653 
.66986 | 1.49284 || .69545 | 1.43792 || .72167 | 1.38568 
67028 | 1.49190 |) .69588 | 1.43703 |) .72211 | 1.38484 
67071 | 1.49097 || .69631 | 1.43614 || .'72255 | 1.88399 
67113 | 1.49008 || .69675 | 1.48525 || .72299 | 1.38314 
.67155 | 1.48909 || .69718 | 1.48486 || .72344 | 1.38229 
67197 | 1.48816 || .69761 | 1.43347 || .72388 | 1.38145 
67239 | 1.48722 || .69804 | 1.43258 || .72432 | 1.38060 
67282 | 1.48629 || .69847 | 1.43169 || .72477 | 1.37976 
67824 | 1.48536 || .69891 | 1.43080 || .72521 | 1.37891 

| *.67366 | 1.48442 || .69934 | 1.42992 || .72565 | 1.37807 
.67409 | 1.48849 ||, .69977 | 1.42903 || .72610 | 1.37722 
| .67451 | 1.48256 || .70021 | 1.42815 || .72654 | 1.87638 

Cotang| Tang ||Cotang| Tang | Cotang| Tang 

56° 55° 54° 


RR A ES 


/ 


© 





= lomworaasza 


TABLE XXVIII.—NATURAL TANGENTS 














; 36° 37° 38° 39° 
Tang | Cotang || Tang | Cotang || Tang | Cotang || Tang | Cotang 
O| .72654 | 1.87638 10855 | 1.32704 78129 | 1.27994 80978 | 1.238490 
1) .72699 | 1.37554 75401 | 1.82624 || .78175 | 1.27917 }| .81027 | 1.238416 
2) .72743 | 1.37470 75447 | 1.382544 || .78222 | 1.27841 81075 | 1.28343 
8) .72788 | 1.37386 75492 | 1.82464 || .78269 | 1.27764 81123 | 1.23270 
4} .72832 | 1.37302 75538 | 1.32384 |] .'78316 | 1.27688 81171 | 1.23196 
5) .72877 | 1.37218 75584 | 1.382804 || .78863 | 1.27611 81220 | 1.23123 
6) _.72921 | 1.37134 75629 | 1.82224 || .78410 | 1.27585 81268 | 1.238050 
7| .72966 | 1.87050 75675 | 1.82144 || .'78457 | 1.27458 81316 | 1.22977 
8} .78010 | 1.36967 75721 | 1.82064 || .'78504 | 1.273882 81364 | 1.22904 
9| .73055 | 1.36883 75767 | 1.381984 || .78551 | 1.27306 81413 | 1.22831 
10} .73100 | 1.36800 || .75812 | 1.31904 || .78598 | 1.27230 81461 | 1.22758 
11| .73144 | 1.36716 75858 | 1.31825 || .78645 | 1.27153 81510 | 1.22685 
12) .738189 | 1.36633 75904 | 1.81745 || .78692 | 1.27077 81558 | 1.22612 
(3) .738234 | 1.36549 75950 | 1.31666 || .78739 | 1.27001 81606 | 1.22539 
14| .73278 | 1.36466 75996 | 1.31586 || .78786 | 1.26925 81655 | 1.22467 
15) .73323 | 1.36383 76042 | 1.31507 || .78834 | 1.26849 81703 | 1.22394 
16) .73368 | 1.36300 76088 | 1.31427 || .'78881 | 1.26774 81752 | 1.22321 
17) .73413 | 1.36217 76184 | 1.31348 || .78928 | 1.26698 81800 | 1.22249 
18) .73457 | 1.36134 76180 | 1.81269 || .78975 | 1.26622 81849 | 1.22176 
19} .73502 | 1.36051 76226 | 1.81190 || .79022 | 1.26546 81898 | 1.22104 
20|-.73547 | 1.35968 76272 | 1.81110 || .'79070 | 1.2647 81946 | 1.22031 
21| .738592 | 1.35865 76318 | 1.31031 79117 | 1.26395 81995 | 1.21959 
22| .738637 | 1.35802 76364 | 1.80952 || .79164 | 1.26319 82044 | 1.21886 
23| .73681 | 1.35719 76410 | 1.30873 || .79212 | 1.26244 82092 | 1.21814 
24| .73726 | 1.35637 76456 | 1.80795 || .79259 | 1.26169 82141 | 1.21742 
20| 738771 | 1.85554 76502 | 1.80716 || .79806 | 1.26093 82190 | 1.21670 
26| .78816 | 1.35472 16548 | 1.30637 || .79854 | 1.26018 82288 | 1.21598 
7| .73861 | 1.35389 76594 | 1.30558 .79401 | 1.259438 82287 | 1.21526 
28| *.73906 | 1.35307 76640 | 1.30480 || .79449 | 1.25867 82386 | 1.21454 
29| .73951 | 1.385224 76686 | 1.30401 .79496 | 1.25792 || .82885 | 1.21382 
30| .73996 | 1.385142 76733 | 1.80823 || .79544 | 1.25717 || .82484 | 1.21310 
31) .74041 | 1.85060 76779 | 1.30244 || .79591 | 1.25642 || .82483 | 1.21238 
32| .74086 | 1.34978 76825 | 1.380166 || .79639 | 1.25567 || .82581 | 1.21166 
33| .74131 | 1.34896 76871 | 1.30087 || .79686 | 1.25492 82580 | 1.21094 
34) .74176 | 1.34814 76918 | 1.30009 || .79734 | 1.25417 || .82629 | 1.21023 
35| .74221 | 1.34782 76964 | 1.29931 || .79781 | 1.25343 || .82678 | 1.20951 
36| .74267 | 1.34650 77010 | 1.29853 || .79829 | 1.25268 || .82727 | 1.20879 
37| .74812 | 1.34568 T7057 | 1.29775 || 279877 | 1.25193 || .82776 | 1.20808 
38| .743857 | 1.34487 77103 | 1.29696 || .79924 | 1.25118 || .82825 | 1.20736 
39| .74402 | 1.34405 77149 | 1.29618 || .'79972 | 1.25044 || .82874 | 1.20665 
40| .74447 | 1.343823 77196 | 1.29541 || .80020 | 1.24969 || .82923 | 1.20593 
41) 74492 | 1,34242 77242 | 1.29463 80067 | 1.24895 || .82972 | 1.20522 
42} .74538 | 1.34160 77289 | 1.29885 80115 | 1.24820 |} .83022 | 1.20451 
43| .74583 | 1.34079 77835 | 1.293807 80163 | 1.24746 || .83071 | 1.20879 
44| .74628 | 1.383998 77382 | -1.29229 80211 | 1.24672 || .83120 | 1.20808 
45| .74674 | 1.33916 77428 | 1.29152 80258 | 1.24597 || .83169 | 1.20237 
46| .74719 | 1.33835 T7475 | 1.29074 80806 | 1.24523 || .88218 | 1.20166 
47| .74764 | 1.38754 77521 | 1.28997 80354 | 1.24449 |] .83268 | 1.20095 
48| .74810 | 1.33673 77568 | 1.28919 80402 | 1.24375 || .88317 | 1.20024 
49) .74855 | 1.33592 77615 | 1.28842 80450 | 1.24801 || .88366 | 1.19958 
50| .74900 | 1.33511 77661 | 1.28764 80498 | 1.24227 || .88415 | 1.19882 
51] .74946 | 1.334380 77708 | 1.28687 80546 | 1.24158 || .83465 | 1.19811 
52| .74991 | 1.33849 77754 | 1.28610 80594 | 1.24079 83514 | 1.19740 
53] .75037 | 1.33268 77801 | 1.28533 80642 | 1.24005 83564 | 1.19669 
54| .75082 | 1.33187 77848 | 1.28456 80690 | 1.23931 836138 | 1.19599 
55| .75128 | 1.33107 77895 | 1.28879 80788 | 1.23858 83662 | 1.19528 
56] .75173 | 1.33026 77941 | 1.28802 80786 | 1.23784 || .88712 | 1.19457 
57} .75219 | 1.32946 77988 | 1.28225 80834 | 1.23710 || .83761 | 1.19387 
58] .75264 | 1.32865 78035 | 1.28148 80882 | 1.23637 || .88811 | 1.19316 
59| .75310 | 1.82785 78082 | 1.28071 80930 | 1.23563 || .83860 | 1.19246 
60) .75355 | 1.32704 78129 | 1.27994 80978 | 1.23490 || .83910 | 1.19175 
, \Cotang| Tang ||\Cotang} Tang |Cotang| Tang | Cotang| Tang 
53° 52° 51° 50° 













































































~ lonwepnanss 





AND COTANGENTS 


199 


























40° 
__|_Tang | Cotang 
0} .83910 | 1.19175 
1} .83960 | 1.19105 
2} .84009 | 1.19035 
3} .84059 | 1.18964 
4| .84108 | 1.18894 
5} .84158 | 1.18824 
6} .84208 | 1.18754. 
7| .84258 | 1.18684 
8} .843807 | 1.18614 
9| .84357 | 1.18544 
10; .84407 | 1.18474 
1i| .84457 | 1.18404 
12} .84507 | 1.18334 
13} .84556 | 1.18264 
14| .84606 | 1.18194 
15| .84656 | 1.18125 
16) .84706 | 1.18055 
17) .84756 | 1.17986 
18| .84806 | 1.17916 
19| .84856 | 1.17846 
20| .84906 | 1.17777 
21} .84956 | 1.17708 
22] .85006 | 1.17638 
23; .85057 | 1.17569 
24| .85107 | 1.17500 
25| .85157 | 1.17430 
26| .85207 | 1.17361 
27| .85257 | 1.17292 
28) .85308 | 1.17223 
29| .85358 | 1.17154 
80} .85408 | 1.17085 
81} .85458 | 1.17016 
22] .85509 | 1.16947 
43] .85559 | 1.16878 
84| .85609 | 1.16809 
35] .85660 | 1.16741 
36} .85710 | 1.16672 
37} .85761 | 1.16603 
88| .85811 | 1.16535 
39| .85862 | 1.16466 
40; .85912 | 1.16398 
41} .85963 | 1.16329 
42) .86014 | 1.16261 
43| .86064 | 1.16192 
44| .86115 | 1.16124 
45| .86166 | 1.16056 
46| .86216 | 1.15987 
47| .86267 | 1.15919 
48! .86318 | 1.15851 
49} .86368 | 1.15783 
50| .86419 | 1.15715 
51| .86470 | 1.15647 
52| .86521 | 1.15579 
53) .86572 |.1.15511 
54| .86623 | 1.15443 
55| .86674 | 1.15375 
56| .86725 | 1.15308 
57| .86776 | 1.15240 
58| .86827 | 1.15172 
59! .86878 | 1.15104 
60} .86929 | 1.15037 
; Cotang| Tang 
49° 




















41° 


Tang | Cotang 








1177 

117138 
11648 
11582 
11517 
11452 
11387 
11321 
.11256 
ed tot 
11126 
11061 


86929 | 1.15037 
86980 | 1.14969 
87031 | 1.14902 
87082 | 1.14834 
87133 | 1.14767 
87184 | 1.14699 
87236 | 1.14632 
87287 | 1.14565 
87338 | 1.14498 
87389 | 1.14430 
87441 | 1.14363 
87492 | 1.14296 
87543 | 1.14229 
87595 | 1.14162 | 
87646 | 1.14095 
87698 | 1.14028 
87749 | 1.13961 
87801 | 1.13894 
87852 | 1.13828 
87904 | 1.13761 
87955 | 1.13694 
88007 | 1.13627 
88059 | 1.13561 
88110 | 1.13494 
88162 | 1.13428 
88214 | 1.13361 
88265 | 1.13295 
88317 | 1.13228 
88369 | 1.13162 
88421 | 1.13096 
88473 | 1.13029 
88524 | 1.12963 
88576 | 1.12897 
88628 | 1.12831 
88680 | 1.12765 
88732 | 1.12699 
88784 | 1.12633 
88836 | 1.12567 
88888 | 1.12501 
88940 | 1.12435 
88992 | 1.12369 
89045 | 1.12308 
89097 | 1.12238 
89149 | 1.12172 
89201 | 1.12106 
89253 | 1.12041 
89306 | 1.11975 
89358 | 1.11909 
so410 | 1.11844 

1 

1 

1 

1 

1 

1 

1 

1 

1 

1 

1 





Cotang| Tang 


48° 














42° 


Tang | Cotang 


11061 


90040 | 4 

-90093 | 1.10996 
90146 | 1.10931 
90199 | 1.10867 
90251 | 1.10802 
90304 | 1.10737 
90357 | 1.10672 
90410 | 1.10607 
90463 | 1.10548 
90516 | 1.10478 
90569 | 1.10414 
90621 | 1.10349 
90674 | 1.10285 
90727 | 1.10220 
90781 | 1.10156 
90834 | 1.10091 
90887 | 1.10027 
90940 | 1.09963 
90993 | 1.09899 
91046 | 1.09834 
91099 | 1.09770 
91153 | 1.09706 
91206 | 1.09642 
91259 | 1.09578 
91313 | 1.09514 
91366 | 1.09450 
91419 | 1.09386 
91473 | 1.09322 
91526 | 1.09258 
91580 | 1.09195 
91633 | 1.09131 
91687 | 1.09067 
91740 | 1.09003 
91794 | 1.08940 
91847 | 1.08876 
91901 | 1.08813 
91955 | 1.08749 
92008 | 1.08686 
92062 | 1.08622 
92116 | 1.08559 
92170 | 1.08496 
92224 | 1.08432 
92277 | 1.08369 
92331 | 1.08306 
92385 | 1.08243 
92439 | 1.08179 
92493 | 1.08116 
92547 | 1.0£053 
-92601 | 1.07990 
92655 | 1.07927 
92709 | 1.07864 
.92763 | 1.07801 
92817 | 1.07738 
92872 | 1.07676 
92926 | 1.07613 
92980 | 1.07550 
93034 | 1.07487 
-93088 | 1.07425 
93143 | 1.07362 
93197 | 1.07299 


93252 | 1.07237 


43° 











. 96064 
.96120 
.96176 
96232 
96288 
. 96344 
.96400 
96457 
.96513 
. 96569 











Cotang| Tang 


47° 








Cotang 


y | Cotang 


1.07237 
1.07174 
1.07112 
1.07049 
1.06987 
1.06925 
1.06862 
1.06800 
1.06738 
1.06676 
1.06613 


1.06551 
1.06489 
1.06427 


—e 
S 
for) 
ey) 
or) 
Or 


Pee ee pe 
(=) 
or 
& 
S) 
nsG 


1.04705 
1.04644 
1.04583 
1.04522 
1.04461 
1.04401 
1.04340 
1.04279 
1.04218 
1.04158 


1.04097 
1.04036 
1.03976 
1.03915 
1.03855 
1.03794 
1.03784 
1.03674 
1.03613 
1.03553 


46° 





@mo 


lomownoaas 


Tang 4 





200. TABLE XXVIII.—NATURAL TANGENTS—COTANGENTS 
rr a 


44° 
Tang | Cotang 


= 








0 | .96569 | 1.03553 
1 | .96625 | 1.03493 
2 | .96681 | 1.03433 
3 | .96738 | 1.03372 
4} .96794 | 1.03312 
5 | .96850 | 1.03252 
6 | .96907 | 1.03192 
7 | .96963 | 1.03132 
8 | .97020 | 1.03072 
9 | .97076 | 1.03012 
10 | .97133 | 1.02952 
11 | .97189 | 1.02892 
12) .97246 | 1.02832 
13 | .973802 | 1.02772 
14 | .97359 | 1.02713 
15 | .97416 | 1.02653 
16 | .97472 | 1.02593 
17 | .97529 | 1.02533 
18 | .97586 | 1.02474 
19 | .97643 | 1.02414 


20 | .97700 | 1.02355 











Cotang Tang 


45° 








, 








44° 
Tang | Cotang 
.97700 | 1.02355 
97756 | 1.02295 
.97813 | 1.02236 
.97870 | 1.02176 
97927 | 1.02117 
.97984 | 1.02057 
98041 | 1.01998 
98098 | 1.01939 
98155 | 1.01879 
98213 | 1.01820 
98270 | 1.01761 
.98327 | 1.01702 
.98384 | 1.01642 
.98441 | 1.01583 
.98499 | 1.01524 
.98556 | 1.01465 
98613 | 1.01406 
98671 | 1.01847 
98728 | 1.01288 
98786 | 1.01229 
98843 | 1.01170 





Cotang| Tang 








45° 























44° 

c 

Tang | Cotang 
98843 | 1.01170 | 20 
98901 | 1.01112 | 19 
98958 | 1.01053 | 18 
.99016 | 1.00994 | 17 
.99073 | 1.00935 | 16 
99131 | 1.00876 | 15 
99189 | 1.00818 | 14 
99247 | 1.00759 | 18 
99304 | 1.00701 | 12 
99362 | 1.00642 | 11 
99420 | 1.00583 | 10 
99478 | 1.00525 | 9 
99536 | 1.00467 | 8 
99594 | 1.00408 | 7 
99652 | 1.00350 | 6 
99710 | 1.00291 | 5 
99768 | 1.00233 | 4 
99826 | 1.00175 | 3 
.99884 | 1.00116 | 2 
.99942 | 1.00058 | 1 
1.00000 | 1.00000 | 0 

Cotang| Tang 





XXIX.—NATURAL VERS. SINES—EXTERNAL SECANTS 201 














Qe 

, 

Vers. |Ex.sec. 

0 | .0000C | .00000 

1 | .00000 / .00000 

2} .00000 | .00000 

8 | .00000 | .00000 

4 | .06000 | .00000 

5 | .00000 |: 700000 

6 | .00000 | ,00000 

7 | .00000 00000 

8 | .00000 | . 

9 | .00000 | .00009 
10 | .00000 | .00000 
11 | .00001 | .00001 
12 | .00001 | .00001 
13 | .00001 | .00001 
14 | .00001 | .00001 
15 |} .00001 | .00001 
16 | .00001 | .00001 

7 | .00001 00001 
18 | .00001 | .00001 
19 | .00002 | .00002 
20 | .00002 | .00002 
21 | .00002 | .00002 
22 | .00002 | .00002 
23 | .00002 | .00002 
24 | .00002 | .00002 
25 | .00003 | .00003 
26 | .00003 | .00003 
27 | .00003 | .00003 
28 | .00003 | .00003 
29 | .00004 | .00004 
30 | .00004 | .00004 
81 | .00004 | .00004 
82 | .00004 | .00004 
33 | .00005 | .00005 
84 | .00005 | .00005 
35 | .00005 | 00005 
36 | 00005 | .00005 
3” 00606 | .00006 
88 | .00006 | .00006 
39 | .00006 | .00006 
40 | .00007 | .00007 
41 00007 | .00007 
42 | .00007 | .00007 
43 | -.00008 | .00008 
44 | .00008 | .00008 
45 | .00009 | .00009 
46 | .00009 | .00009 
47 | .00069 | .00009 
48 | .00010 00010 
49 | .0001 00010 
50 | .00011 00011 
51 | .00011 00011 
52 | .00011 00011 
53 | .00012 | .00012 
54 | .00012 | .00012 
55 | .00013 | .00013° 
56 | .00013 | .00013 
57 | .00014 | .00014 
58 | .00014 | .00014 
59 | .00015 | .00015 
60 | .00015 | . 

















1° 


Vers. |Ex. sec. 


00015 | .00015 
00016 | .00016 
00016 | .00016 
00017 | .00017 
00017 | .00017 
00018 | .00018 
00018 | .00018 
00019 | .00019 
00020 | .00020 
00020 | .00020 
00021 | .00021 
00021 | .Q0021 


00023 | .00023 
00024 | .00024 
00024 | .Q0024 
00025 | .00025 


00027 | .00027 
00028 | .00028 
00028 | .00028 
0002 0002 
00030 | .00030 
00031 | .00031 
00031 | .00031 


00041 | .00041 
00041 | .00041 
00042°| .00042 
00043 | .00043 


.00044 | .00044 
.00045 | .00045 


-00046 | .00046 
.00047 | .00047 
.00048 | .00048 
.00048 | .00048 
-00049 | .00049 
-00050 | .00050 


.00051 | .00051 
.00052 | .00052 


.00053 | .00053 
00054 | .00054 
00055 | .00055 
00056 | .00056 
00057 | .00057 
00058 | .00058 
00059 | .00059 
00060 | .00060 
00061 | .00061 











9° 





Vers. |Ex.sec. 


.00061 
.00062 
.00063 
00064 
00065 
.00066 
00067 
.00068 
.00069 
.00070 
90072 


00073 
00074 
00075 
.00076 
00077 
.00078 
.00079 
.00081 
.00082 
.00083 


00084 
00085 
00087 
00088 
.00089 
.00090 
.00091 
.00093 
00094 
.00095 


.00097 
.00098 
.00099 
.00100 
00102 
00103 
00104 
00106 
00107 
.00108 


-00110 
00111 
00118 
00114 
00115 
.00117 
.00118 
.00120 
00121 
00122 


00124 
00125 
00127 
.00128 
.00130 
00131 
.00133 
.00134 
-00136 
.00137 





Vers. |Ex. sec. 


8° 


—_—— —— | —— | 


.00137 











.00137 


00139 


.00140 
.00142 
.00143 
00145 
00147 
.00148 
.00150 
.00151 
.00153 


.00155 
.00156 
.00158 
.00159 
.00161 
.00163 
.00164 
.00166 
.00168 
.00169 


00171 
00173 
.00175 
.00176 
.00178 
.00180 
00182 
00183 
.00185 
00187 


00189 





OMI OUP CWE © 


10 





202 





= 


{ 








TABLE XXIX.—NATURAL VERSED SINES 


























| 








go 5° 

Vers. |Ex.sec.|| Vers. 'Ex. sec. 
.00244 | .00244 00381 | .00382 
.00246 | .00246 00383 | .00385 
.00248 | .00248 00386 | .00387 
.00250 | :00250 00388 | .00390 
.00252 | .00252 00391 | .00392 
.00254 | .00254 00393 | .00395 
.00256 | .00257 00396 | .00897 
.00258 | .00259 00398 | .00400 
.00260 | .00261 00401 | .00403 
.00262 | .00263 00404 | 00405 
.00264 | .00265 || .00406 | .00408 
.00266 | .00267 || .00409 | .00411 
.00269 | .00269 |} .00412 | .00413 
.00271 00271 .00414 | .00416 
.002738 | .00274 00417 | .00419 
.00275 | .00276 00420 | .00421 
0027 00278 00422 | .00424 
.00279 | .00280 00425 | .00427 
.00281 00282 00428 | .00429 
.00284 | .00284 || .00430 | .00432 
.00286 | .00287 || .00433 | .00435 
.00288 | .00289 || .00436 | .00438 
.00290 | .00291 || .00438 | .00440 
00298 | .00293 00441 | .00443 
00295 | .00296 00444 | .00446 
00297 | .00298 00447 | .00449 
00299 | .00300 00449 | .00451 
00301 00302 00452. | .00454 
00304 | .00305 00455 | .00457 
00306 | .00307 00458 | .00460 
00308 | .00309 00460 | .00463 
00311 00312 00463 | .00465 
00313 | .00814 || .00466 | .00468 
00315 | .00816 || .00469 | .00471 
00317 | .00818 || .00472 | .0047 

00320 | .00321 .00474 | .00477 
00822 | .00823 || .00477 | .00480 
00324 | .00326 || .00480 | .00482 
00827 | .003828 || .00483 | .00485 
00329 | .00380 |} .00486 | .00488 
00332 | .00333 00489 | .00491 
00334 | .00335 00492 | .00494 
00336 | .00337 00494 | .00497 
00339 | 00340 00497 | .00500 
00341 | .00342 00500 | .00503 
00843 | .00845 00503 | .00506 
00346 | .00347 00506 | .00509 
00348 | .00350 00509 | .00512 
00351 | .00352 00512 | .00515 
00353 | .00354 00515 | .00518 
00356 | .00357 00518 | .00521 
00358 | .00359 00521 00524 
00361 00362 00524 | .00527 
00363 | 00864 00527 | .00530 
00365 | .00367 00580 | .00533 
00368 | .00269 00533 | .00536 
00370 | .0037 00536 | .00539 
00373 | .00374 00539 | .00542 
-00875 | 200377 00542 | .00545 
-00878 | .90379 00545 | .00548 
.00381 | .00382 00548 | .00551 











Vers. |x. sec. 


.00551 
00554 
00557 
.00560 
00563 
00566 
00569 
00573 
00576 
00579 
00582 


00585 
.00588 
00592 
00595 


.00548 
£00551 
.00554 
.00557 
00560 
00563 
00566 
00569 
00572 
.00576 
00579 


. 00582 
00585 
00588 
- 00591 
00594 
-00598 
00601 
00604 
00607 
.00610 


.00614 
.00617 
.00620 
.00623 
. 00626 
.00630 
00633 
00636 
00640 
00643 


00646 
00649 
.00653 
.00656 
.00659 


6° 


. .00598 





.00601 
00604 
.00608 
00611 
00614 


.00617 
00621 
00624 
00627 
.00630 
00634 
00637 
.00640 
00644 
00647 


00650 
00654 
00657 
.00660 
00664 
.00667 
00671 
00674 
00677 
00681 


00684 
00688 
00691 
00695 
00698 
00701 
00705 
00708 














vhs 


~ 


Vers. |Ex. sec. 


00745 | 00751 | 0 
00749 | .00755 | 1 
00752 | .00758 | 2 
00756 | .00762 | 3 
00760 | .00765 | 4 © 
00763 | .00769 | 5 
00767 | .00773 | 6 
00770 | .00776 | 7 
00774 | .00780 | 8 
00778 | .00784 | 9 
00781 | .00787 | 10 
00785 | .00791 | 11 
00789 | .00795 | 12 
00792 | .00799 | 13. 
00796 | .00802 | 14 


“00800 | .00806 | 15 


(oe: 
S 
Re) 
<2) 
on 
ea 
So 
a) 
(oC) 
as 
a 
le 3) 








AND EXTERNAL SECANTS 





0 | .00973 | .00983 
1 | .00977 | .00987 
2| .00981 | .00991 
3 | .00985 | .00995 
4 | .00989 | .00999 
5 | .00994 | .01004 
6 | .00998 | .01008 
7 | .01002 | .01012 
8 | .01006 , .01016 
9 | .01010 | .01020 
10 | .01014 | .01024 


11 | .01018 | .01029 
12 | .01022 | .01033 
13 | .01027 | .01037 
14 | .01031-; .01041 
15 | .01035 | -.01046 
16 | .01039 | .01050 
17 | .01043 | .01054 
18 | .01047 | .01059 
19 | .01052 | .01063 
20 | .01056 | .01067 


21 | .01060 | .01071 





22 | .01064 | .01076 |. 


23 | .01069 | .01080 
24 | .01073 | .01084 
25 | .01077 | .01089 
26 | .01081 | .01093 
27 | .01086 | .01097 
28 | .01090 | .01102 
29 | .01094 | .01106 
30 | .01098 | .01111 


31 | .01103 ; .01115 
32 | .01107 | .01119 
33°} .O1111 | .01124 
34 | .01116 | .01128 
385 | .01120 | .01133 
36 | .01124 | .01137 
3¢ | .01129 | .01142 
38 | .01183 | .01146 
39 | .01187 | .01151 
40 | .01142 | .01155 


41 | .01146 | .01160 
42 | .01151 | .01164 
43 | .01155 | .01169 
44 | .01159 | .01173 
45 | .01164 | .01178 
46 | .01168 | .01182 
47 | .01173 | .01187 
48 | .01177 ; .01191 
49 | .01182 | .01196 
50 | .01186 | .01200 


51 | .01191 | .01205 
52 | .01195 | .01209 
53 | .01200 | .01214 





—-B4 |} .01204 | .01219 


55 | .01209 | .01223 
56 | .01213 | .01228 
57 | .01218 | .01233 
58 | .01222 | .01237 
59 | .01227 | .01242 
60 | .01231 | .01247 














9° 10° 

Vers. |Ex.sec.|| Vers. Lae. sec. 
.01231 01247 01519 | .01543 
.01286 01251 01524 | .01548 
.01240 01256 01529 | .01553 
.01245 | .01261 015384 | .01558 
.01249 | .01265 01540 | .01564 
.01254 | .0127 01545 | .01569 
.01259 | .01275 01550 | .01574 
.01263 01279 01555 | .01579 
.01268 01284 01560 | .01585 
.01272 01289 01565 | .01590 
.01277 01294 01570 | .01595 
.01282 | .01298 || .015% .01601 
.01286 01303 .01580 | .01606 
.01291 01308 01586 | .01611 
.01296 013138 01591 | .01616 
.01300 01318 01596 | .01622 
.01805 01822 01601 | .01627 
.01819 01327 01606 | .016383 
013814 013382 01612 | .01638 
.01319 013387 01617 | .01643 
.01824 01342 01622 | .01649 
.01329 01346 .01627 | .01654 
.01333 | .01351 -01632 | .01659 
-01388 | .01356 .01688 | .01665 
.01343 |. .01361 .01643 | .01670 
.01348 | .01366 .01648 | .01676 
.01352 | .0137 .01653 | .01681 
.01357 | .013876 || .01659 | .01687 
.01362 01381 .01664 | .01692 
.01367 01386 .01669 | .01698 
.013871 01391 .01675 | .017038 
.01376 | .01895 || .01680 | .01709 
.01381 01400 .01685 | .01714 
.01386 01405 .01690 | .01720 
.01391 | .01410 ||} .01696 | .01725 
.01396 01415 .01701 | .01731 
.01400 01420 01706 | .01736 
.01405 01425 .01712 | .01742 
01410 01430 01717 | .01747 
01445 | .01435 .01723 | .01753 
-01420 01440 .01728 | .01758 
.01425 | .01445 || .01783 | .01764 
.01439 01450 .01739 | .01769 
-01435 01455 01744 | 01775 
.01489 01461 .01750 | .01781 
.01444 01466 .01755 | .01786 
01449 01471 .01760 | .01792 | 
.01454 01476 .01766 | .01798 
.01459 01481 01771 | .01803 
.01464 01486 01777 | .01809 
.01469 01491 01782 | .01815 
.01474 01496 01788 | .01820 
.01479 01501 01793 | .01826 
.01484 01506 01799 | .01882 
101489 01512 01804 | .01837 
.01494 01517 01810 | .01843 
01499 01522 01815 | .01849 
.01504 01527 01821 | .01854 
-01509 | .01532 01826 | .01860 
.01514 | .01537 01832 | .01866 
.01519 | .01543 01887 | .01872 






































17 


11° 
Vers. |Ex. sec. 
018387 | .01872 
01843 | .01877 
01848 | .01883 
01854 | .01889 
01860 | .01895 
01865 | .01901 
01871 | .01906 
01876 | .01912 
01882 | .01918 
01888 | .01924 
01893 | .01930 
01899 | .01936 
01904 | .01941 
01910 | .01947 
01916 | .01953 
01921 | .01959 
01927 | .01965 
01983 | .01971 
01939 | .01977° 
01944 | .019838 
01950 | .01989 
01956 |} .01995 
01961 | .02001 
01967 | .02007 
01973 | .020138 
01979 | .02019 
01984 | .02025 
01990 | .02031 
01996 | .02037 
.02902 | .02043 
.02008 | .02049 
.02013 | .02055 
02019 | .02061 
.C2025 | .02067 
.02031 | .02073 
.02087 | .02079 
.02042 | .02085 
.02048 | .02091 
,02054 | .02097 
.02050 | .02103 
.02066 | .02110 
.02072 | .02116 
-02078 | .02122 
.02084 | .02128 
.02090 | .02134 
.02095 | .02140 
.02101 | .02146 | 
.02107 | .021538 
.02113 | .02159 
.02119 | .02165 
02125 | .02171 
.02131 | .02178 
.02137 | .02184 
.02143 | .02190 
,02149 | .02196 
.02155 | 02203 
.02161 | .02209 
02167 | .02215 
.02173 | .02221 
.02179 | .02228 
.02185 | .022384 











—_— re SS 


204 TABLE XXIX.—NATURAL VERSED SINES 





12° 18°. 14° 15° 


Vers. |Ex.sec.;| Vers. |Ex.sec.|} Vers. |Ex.sec.|] Vers. |Ex. sec. 


—_——_—— | | | | | | | |S | | | | 


.02185 | .02284 || .02563 | .02630 || .02970 | .03061 || .03407 | .03528 
.02191 | .02240 || .02570 | .02637 || .02977 | .08069 || .03415 | .03536 
.02197 | .02247 || .02576 | .02644 || .02985 | .03076 || .03422 | .08544 
.02203 | .02253 || .02583 | .02651 || .02992 | .038084 || .03480 | .03552 
.02210 | .02259 || .02589 | .02658 ||} .02999 | .03091 || .03438 | .03560 
.02216 | .02266 || .02596 | .02665 |} .08006 | .03099 || .03445 | .03568 
02222 | .02272 || .02602 | .02672 || .03013 | .03106 || .03453 | .03576 
02228 | .02279 || .02609 | .02679 || .08020 | .08114 || .03460 | .03584 
.02234 | .02285 || .02616 | .02686 || .03027 | .03121 |} .08468 | .03592° 
02240 | .02291 || .02622 | .02693 || .03034 | .03129 || .03476 | .03601 
02246 | .02298 || .02629 | .02700 || .03041 | .03187 || .08483 | .08609 


11 | .02252 | .02304 || .02635 | .02707 || .03048 | .08144 || .03491 | .08617 | 11 
12 | .02258 | .02311 |) .02642 | .02714 || .08055 | .03152 || .03498 | .08€25 | 12 
13 | .02265 | .02317 || .02649 | .02721 || .03063 | .08159 || .038506 | .08633 | 18 
14 | .02271 | .02823 || .02655 | .02728 || .08070 | .08167 || .03514 | .038642 | 14 
15 | .02277 | .02880 || .02662 | .02735 || .038077 | .08175 || .08521 | .08650 | 15 
16 | .02283 | .02336 || .02669 | .02742 || .03084 | .03182 || .03529 | .03658 | 16 
17 | .02289 | .02343 || .02675 | .02749 || .08091 | .03190 || .03537 | .03666 | 17. 
18 | .02295 | .02349 || .02682 | .02756 || .03098 | .03198 || .03544 | .03674 | 18 
19 | .02302 | .02356 || .02689 | .02763 |} .03106 | .08205 || .03552 | .03683 | 19 
20 | .023808 | .02362 || .02696 | .0277 -03118 | .038213 || .03560 | .08691 | 20 


21 | .02314 | .02369 || .02702 | .02777 || .038120 | .03221 || .03567 | .038699 | 21 
22 | .02820 | .02375 || .02709 | .02784 || .03127 | .03228 || .03575 | .03708 | 22 
23 | .02827 | .023882 || .02716 | .02791 || .03184 | .03286 || .03583 | .03716 | 23 
24 | .023833 | .02388 || .02722 | .02799 || .03142 | .03244 || .03590 | .038724 | 24 
25 | .02839 | .02395 || .02729 | .02806 || .03149 | .03251 || .03598 | .037382 | 25 
26 | .02845 | .02402 || .027386 | .02818 || .08156 | .03259 || .03606 | .038741 | 26 
27 | .02852 | .02408 |} .02743 | .02820 || .03163 | .03267 || .03614 | .03749 | 27 
28 | .02858 | .02415 || .02749 | .02827 || .038171 | .03275 || .03621 | .03758 | 28 
29 | .02364 | .02421 || .02756 | .02834 || .03178 | .03282 || .03629 | .03766 | 29 
80 | .02870 | .02428 || .02763 | .02842 |} .03185 | .03290 || .03687 | .03774 | 30 


31 | .02877 | .02435 || .02770 | .02849 || .03193 | .03298 || .03645 | .03783 | 31 
32 | .023883 | .02441 || .02777 | .02856 || .03200 | .03306 || .03653 | .03791 | 32 © 
33 | .023889 | .02448 || .02783 | .02863 || .08207 | .03313 || .03660 | .03799 | 33 
84 | .023896 | .02454 |) .02790 | .0287' -03214 | .03821 || .03668 | .03808 | 34 
85 | .02402 | .02461 || .02797 | .02878 || .03222 | .03329 || .03676 | .03816 | 35 
36 | .02408 | .02468 || .02804 | .02885 || .03229 | .03337 || .03684 | .03825 | 36 
7 | .02415 | .02474 || .02811 | .02892 || .03236 | .03345 |) .03692 | .03833 | 387 — 
38 | .02421 | .02481 || .02818 | .02899 || .03244 | .03353 || .03699 | .03842 | 38 — 
4 








SomaankwmwHe | 
ee 
COMDVHOUP WMH OS 


























39 | .02427 | .02488 || .02824 | .02907 |} .08251 | .03360 || .03707 | .03850 | 39 ~ 
40 | .02434 | .02494 |} .02831 | .02914 || .038258 | .03368 || .03715 | .03858 | 40 — 


41 | .02440 | .02501 |) .02888 | .02921 || .08266 | .03376 || .03723 | .03867 | 41 — 
42 | .02447 | .02508 || .02845 | .02928 || .03273 | .03384 || .03731 | .03875 | 42 — 
43 | .02453 | .02515 || .02852 | .02986 || .03281 | .03392 || .03739 | .03884 | 43 — 
44 | .02459 | .02521 |} .02859 | .02943 || .03288 | .03400 || .03747 | .03892 | 44 i 
45 | .02466 | .02528 |} .02866 | .02950 |} .03295 | .03408 || .03754 | .03901 | 45 — 
46 | .02472 | .02535 || .02873 | .02958 |) .03303 | .03416 || .03762 | .03909 | 46 








47 | .02479 | .02542 || .02880 | .02965 || .03810 | .03424 || .08770 | .03918 | 47 

48 | .02485 | .02548 || .02887 | .0297 .03318 | .03432 || .03778 | .03927 | 48- 
49 | .02492 | .02555 |} .02894 | .02980 || .03825 | .03439 || .03786 | .03935 | 49 — 
50 | .02498 | .02562 || .02900 | .02987 || .03383 | .03447 || .038794 | .03944 | 50 


51 | .02504 | .02569 || .02907 | .02994 || .03340 | .03455 || .03802 | .03952 | 51 
52 | .02511 | .02576 || .02914 | .03002 || .03347 | .03463 || .03810 | .03961 | 52° 
53 | .02517 | .02582 || .02921 | .03009 || .03355 | .03471 || .03818 | .03969 | 53 
54 | .02524 | .02589 || .02928 | .03017 || .03362 | .0347 .03826 | .03978 | 54 
55 | .02530 | .02596 ||} .02935 | .03024 || .03370 | .08487 || .03834 | .03987 | 55 
56 | .02537 | .02603 || .02942 | .03032 || .03377 | .08495 || .03842 | .03995 | 56 
57 | .02548 | .02610 |} .02949 | .03039 || .03385 | .03503 || .08850 | .04004 | 57 
58 | .02550 | .02617 || .02956 | .03046 || .033892 | .03512 || .03858 | .04013 | 58- 
59 | .02556 | .02624 || .02963 | .03054 || .03400 | .03520 || .08866 | .04021 | 59 
60 | .02563 | .02630 || .02970 | .03061 || .03407 | .03528 || .03874 | .04030 | 60 





























AND EXTERNAL SECANTS 205 
16° 17° 18° 19° 
9 2 
Vers. |Ex.sec.!| Vers. |Ex.sec.|| Vers. |Ex.sec.|| Vers. |Ex. sec. 
0 | .03874 | .04030 .04370 | .04569 04894 | .05146 || .05448 | .05762 0 
1} .08882 | .04039 .04378 | .04578 04903 | .05156 || .05458 | .05773 1 
2} .038890 | .04047 04887 | .04588 04912 | .05166 05467 | .057838 2 
3 | .08898 | .04056 04395 | .04597 04921 | .05176 || .05477 | .05794 3 
4 | .08906 | .04065 .04404 | .04606 04980 | .05186 05486 | .05805 4 
5 | .03914 | .04073 .04412 | .04616 04989 | .05196 05496 | .05815 5 
6 | .03922 | .04082 .04421 | .04625 04948-| .05206 || .05505 | .05826 6 
7 | 03980 | .04091 .04429 |, .04685 04957 | .05216 | 105515 .05836 wy 
8 | .039388 | .04100 .04488 | .04644 04967 | .05226 05524 | .05847 8 
9 | .03946 | .04108 -04446 | .04653 04976 | .05236 || .05534 | .05858 9 | 
10 | ..08954 | .04117 .04455 | .04663 04985 | .05246 || .05548 | .05869 | 10 
11 | .03963 | .04126 || .04464 | .04672 04994 | .05256 05553-| .05879 | 11 
12 |. .03971 | .04135 || .04472 | .04682 050038 | .05266 05562 | .05890 | 12 
13 | .08979 | .04144 || .04481 | .04691 05012 | .05276 || .05572 | .05901 | 13 
14 | .038987 .04152 || .04489 | .04700 05021 | .05286 || .05582 | .05911 | 14 
15 | .03995 | .04161 || .04498 | .04710 .05080 | .05297 05591 | .05922 | 15 
16 | .04003 1 .04170 || .04507 | .04719 . 05038 .05807 || .05601 | .05933 | 16 
17 | .04011 | .04179 || (04515 | .04729 05048 | .05817 05610 | .05944 | 17 
18 | .04019 | .04188 04524 | .04738 .05057 | .053827 || .05620 | .05955 | 18 
19 | .04028 | .04197 | .04533 | .04748 05067 | 05837 05630 | .05965 | 19 
20 | .04036 | .04206 | .04541 | .04757 .0507 .05347 05639 | .05976 | 20 
21 | .04044 |_ 04214 || .04550 | .04767 05085 | .05357 05649 | 05987 | 21 
22 | .04052 | .04223 .04559 | .0477 05094 | .05367 05658 | .05998 | 22 
23 | .04060 |, .04282 - .04567 | .04786 05103 | .05378 05668 | .06009 | 23 
24 | .04069 | .04241 04576 | .04795 05112 | .05888 | .05678 | .06020 | 24 
25 | .04077 | .04250 04585 | .04805 05122 | .05398 || .05687 | .06030 | 25 
26 | .04085 | .04259 .04593 | .04815 05131 | .05408 || .05697 | .06041 | 26 
” | 04093 | .04268 .04602 | .04824 05140 | .05418 || .05707 | .06052 | 27 
98 | .04102 | .04277 .04611 | .04834 05149 | .05429 05716 | .06068 | 28 
29 | .04110 | .04286 .04620 | .04843 05158 | .05489 05726 | .06074 | 29 
80 | .04118 | .04295 .04628 | .04858 05168 | .05449 || .057386 | .06085 | 30 
21 | .04126 | .043804 || -.04637 | .04863 05177 | .05460 || .05746 | .06096 | 31 
82 | .04185 | .04313 04646 | .04872 05186 | .05470 || .05755 | .06107 | 32 
83 | .041438 | .04322 .04655 | .04882 05195 | .05480 || .05765 | .06118 | 33 
84 | .04151 | .04831 .04663 | .04891 05205 | .05490 || .05775 | .06129 | 34 
85 | .04159 | .04840 .04672 | .04901 05214 | .05501 || .05785 | .06140 | 35 
36 | .04168 | .04349. 04681 | .04911 05223 | .05511 05794 | .06151 | 86 
87 | .04176 | .04358 .04690 | .04920 05232 | .05521 05804 | .06162 | 37 
88 | .04184 | .04367 || .04699 | .04930 05242 | .055382 05814 | .06173 | 38 
89 | .04193 | .04376 .04707 | .04940 05251 | .05542 05824 | .06184 | 39 
40 | .04201 | .04885 .04716 | .04950 05260 | .05552 05833 | .06195 | 40 
41 | .04209 | .04894 || .04725 | .04959 || .0527' .05563 || .05848 | .06206 | 41 
42 | .04218 | .04403 || .04734 | .04969 0527 .05573 .05858 | .06217 | 42 
43 | .04226 |} .04413 .04743 | .04979 05288 | .05584. .05863 | .06228 | 43 
44 | .04234 | 104422 .04752 | .04989 05298 | .05594 || .0587 .06239 | 44 
45 | .042438 | .04431 .04760 | .04998 05307 | .05604 || .05882 | .06250 | 45 
46 | .04251 | .04440 .04769 | .05008 05316 | .05615 || .05892 | .06261 | 46 
47 | .04260 | .04449 0477 .05018 05326 | .05625 || .05902 | .06272 ve 
48 | .04268 | .04458 .04787 | .05028 05385 | .05636 || .05912 | .06283 | 48 
49 | .04276 | .04468 .04796 | .05038 05344 | .05646 | .05922 |. .06295 | 49 
50 | .04285 | .04477 .04805 | .05047 05354 | .05657 05932 | .06806 | 50 
51 | .04293 | .04486 04814 | .05057 || .05363 | .05667 05942 | .06317 | 51 
52 | .04802 | .04495 .04823 | .05067 .05373 | .05678 05951 | .06328 | 52 ~ 
53 | .04810 | .04504 .04882 | .05077 05382 | .05688 05961 | .06839 | 53 
54 | .04319 | .04514 .04841 | .05087 .05391 | .05699 .05971 | .06850 | 54 
55 | .04827 | .04523 || 104850 | .05097 .05401 | .05709 .05981 | .06362 | 55 
56 | .04336 | .04532 .04858 | .05107 .05410 | .05720 05991 | .06873 | 56 
57 | .04844 | .04541 .04867 | .05116 .05420 | .05730 06001 | .06384 | 57 
58 | .04853 | .04551 0487 .05126 .05429 | .05741 .06011 | .06395 | 58 
59 | .04861 | .04560 .04885 | .05136 .05489 | .05751 .06021 |: .06407 | 59 
60 | .04370 | .04569 04894 | .05146 .05448 | .05762 .06031 | .06418 | 60 


SS errr ee 


























“ 





























206 


TABLE XXIX.—NATURAL VERSED SINES 








— 
See eros | 








20° 

Vers. |Ex. sec 
06031 | .06418 
06041 | .06429 
.06051 | .06440 
.06061 | .06452 
.06071 | .06463 
06081 | .06474 
06091 | .06486 
06101 | .06497 
06111 | .06508 
06121 | .06520 
06131 | .06531 
06141 | .06542 
06151 06554 
06161 | .06565 
06171 | .06577 
06181 | .06588 
.06191 | .06600 
.06201 06611 
.06211 06622 
.06221 | .06634 
.06231 06645 
.06241 | .06657 
. 06252 06668 
.06262 | .06680 
.06272 | .06691 
.06282 | .06703 
.06292 | .06715 
.06302 | .06726 
06312 | .06738 
06323 | .06749 
06333 | .06761 
06348 | .06773 
063538 | .06784 
06363 | .06796 
06374 | .06807 
06384 | .06819 
06394 | .06831 
06404 | .06843 
06415 | .06854 
06425 | .06866 
06485 | .0687 

06445 | .06889 
06456 | .06901 
06466 06913 
06476 | .06925 
06486 | .06936 
06497 | .06948 
06507 | .06960 
06517 | .06972 
06528 | .06984 
06538 | .06995 
06548 | .07007 
06559 07019 
06569 | .07031 
06580 | .07043 
06590 | .07055 
06600 | .07067 
06611 079 
06621 07091 
06632 | .07103 
06642 





21° 


Vers. |Ex. sec. 


.06642 
.06652 
.06663 
.06673 
.06684 
.06694 
06705 
06715 
.06726 
06736 
06747 


06757 
06768 
06778 
06789 
06799 
.06810 
-06820 
.06831 
.06841 
06852 


.06863 
06873 





06884 
.06894 
.06905 
.06916 
.06926 
06937 
.06948 
.06958 


.06969 
.06980 
.06990 
07001 
.07012 
07022 
07033 








07044 
07055 
07065 


, 07076 
07087 
-07098 
.07108 
07119 
.07130 
.07141 
07151 
.07162 
07173 


07184 
07195 
07206 
07216 
07227 
07238 
07249 
07260 
07271 

07282 


07115 
07126 
07188 
07150 
07162 
07174 
.07186 
.07199 
07211 
07223 
07235 


07247 
07259 
07271 
07283 
07295 
07307 
.073820 
07332 
07344 
07356 


07368 
07380 
.07393 
07405 
07417 
07429 
07442 
07454 
07466 
07479 


07491 
07503 
.07516 
07528 
07540 
07553, 
07565 
07578 
07590 
07602 


07615 
07627 
.07640 
.07652 
.07665 
07677 
.07690 
07702 
07715 
07727 


07740 
07752 
07765 
07778 
07790 
07803 
07816 
07828 
07841 
07853 























22° 

Vers. |Ex. sec. 
07282 | .07853 
07298 | .07866 
07303 | .07879 
07314 | .07892 
073825 | .07904 
07386 | .07917 
07847 | .07980 
07358 | .079438 
07369 | .07955 
07380 | .07968 
07391 | .07981 
07402 | .07994 
07413 | .08006 
07424 | .08019 
07485 | .08032 
07446 | .08045 
.07457 | .08058 
.07468 | .08071 
0747 . 08084 
.07490 | .08097 
07501 | .08109 
07512 |} .08122 
07523 | .08135 
07534 | .08148 
07545 | .08161 
07556 | .08174 
07568 | .08187 
07579 | .08200 
07590 | .08213 
07601 | .08226 
07612 | .08239 
07623 | .08252 
07634 | .08265 
07645 | .08278 
07657 | .08291 
.07668 | .08305 
.07679 | .083818 
07690 | .08831 
07701 | .08344 

7713 | .083857 
07724 | .083870 
07735 | .083883 
07746 | .08397 
07757 | .08410 
07769 | .08423 
07780 | .084386 
07791 08449 
07802 | .08463 
07814 | .08476 
07825 | .08489 
07886 | .08503 
07848 | .08516 
.07859 | .08529 
.07870 | .08542 
07881 08556 
07893 | .08569 
07904 | .08582 
07915 | .08596 
07927 | .08609 
07988 | .08623 
07950 | .08636 
































23° 
Vers. |Ex. sec. 
07950 | .08636 
07961 | .08649 
07972 | .08663 
07984 | 08676 
07995 | .08690 
08006 | .08703 
08018 | .08717 
08029 | .08730 
08041 | .08744 
08052 | .08757 
08064 | .087'71 
0807 .08784 
.08086 | .08798 
08098 | .08811 
08109 | .08825 
08121 | .08839 
08182 | .08852 
08144 | .08866 
08155 | .08880 
08167 | .08893 
08178 | .08907 
08190 | .08921 
08201 08934 
08213 | .08948 
08225 | .08962 
08236 | .08975 
08248 | .08989 
08259 | .08003 
08271 09017 
08282 | .09080 
08294 | .09044 
08306 | .09058 
08317 | .09072 
.08829 | .09086 
.083840 | .09099 
.08852 | .09113 
.083864 | .09127 
.08375 | .09141 
.08887 | .09155 
.08399 | .09169 
.08410 | .09188 
.08422 | .Q9197 
.08434 | .09211 
.08445 | .09224 
.08457 | .09288 
.08469 | 09252 
08481 09266 
.08492 | .09280 
08504 | .09294 
08516 | .09808 
08528 | 093828 
08539 | .09337 
.08551 09351 
.08563 | .09365 
.08575 | .0937' 
.08586 | .09393 
.08598 | .09407 
08610 | .09421 
08622 | .09435 
08634 | ,09449 
08645 | .09464 














AND EXTERNAL SECANTS 





sip te eset aye | 








24° 

Vers, |Ex. sec 
08645 | .09464 
“08657 | .09478 
“08669 | .09492 
-W8681' | 09506 
-08693 | .09520 
08705 | .09535 
08717 | .09549 
“08728 | 09563 
08740 | .0957 

08752 | .09592 
08764 | .09606 
08776 | .09620 
“08788 | .09635 
“08800 | .09649 
“08812 | .09663 
“08824 | .09678 
“08836. } .09692 
“08848 | .09707 
“08360 | .09721 
‘08872 | .09735 
“08884 | .09750 
08896 | .09764 
“08908 | .0977 

"08920 | .09793 
"08932 | :09808 
"08944 | .09822 
“08956 | .09837 
"08968 | .09851 
"08980 | .09866 
“08992 | .09880 
"09004 | .09895 
.09016 | .09909 
“09028 | .09924 
“09040 | .09939 
“09052 | .09953 
"09064 | .09968 
“09076 | .09982 
“09089 | .09997 
"09101 | .10012 
“09113 | .10026 
(09125 | .10041 
.09137 | .10055 
"09149 | .10071 
‘09161 | .10085 
09174 | .10100 
-09186 | .10115 
‘09198 | .10130 
09210 | .10144 
"09222 | .10159 
"09234 | .101'74 
09247 | .10189 
00259 | .10204 
09271 | .10218 
"09283 | .10233 
09296 | 10248 
09308 | .10263 
09320 | .1027 

09332 | .10293 
09345 | .10308 
09357 | .10323 
09369 | .10338 




















25° 


Vers. | Ex. sec. 


.09369 | .10388 
.09382 | .10853 
.09394 | .10368 
.09406 | .10383 
.09418 | .10398 


.09431 | .10418 
09443 | .10428 
.09455 | .10443 


.09468 | .10458 
.09480 | .10473 
.09493 | .10488 


.09505 | .10503 
.09517 | .10518 
.09580 | .10533 
.09542 | .10549 
.09554 | .10564 
.09567 | .10579 
.09579 | 10594 
.09592 | .10609 
.09604 | .10625 
.09617 | .10640 


.09629 | .10655 
.09642 | .10670 
.09654 | .10686 
.09666 | .10701 
-09679 | .10716 
.09691 | .10781 
.09704 | .10747 
.09716 | .10762 
09729 | .10777 
.09741 | .10793 
.09754 | .10808 
09767 | .10824 
.09779 | .10839 
‘09792 | .10854 
.09804 | .10870 
.09817 | .10885 
.09829 | .10901 
.09842 | .10916 
.09854 | .10982 
.09867 | .10947 


.09880 | .10963 
.09892 | .10978 
.09905 | .10994 
.09918 | .11009 
.09930 | .11025 
.09943 | .11041 
.09955 | .11056 
.09968 | 11072 
.09981 | .11087 
09993 | .11103 


-10006 | .11119 
.10019 | .11134 


-10082 | .11150 © 


.10044 | .11166 
510057 | .11181 
.10070 | 11197 
10082 | .11213 
.10095 | .11229 
.10108 | .11244 
.10121 | .11260 











26° 


10121 
"10133 
110146 
“10159 
10172 
110184 
10197 
10210 
10223 
10236 
10248 


10261 
10274 
10287 
. 10300 
.103138 
. 10326 
10838 
10851 
10364. 
10377 


. 10390 
10403 
.10416 
. 10429 
.10442 
. 10455 
10468 
10481 
10494 
. 10507 


10520 
10533 
.10546 
. 10559 
10572 
10585 
.10598 
.10611 
10624 
. 10687 


. 10650 
-10663 
. 10676 
. 10689 
10702 
10715 
10728 
10741 
10755 
10768 


10781 
10794 
10807 
. 10820 
10833 
10847 
.10860 
10873 
. 10886 
10899 











Vers. |Ex. sec. 


“12067 


12084 
12100 
aleilaure 
12133 
12150 
12166 
12183 
.12199 
12216 
. 122338 











27 


Vers. |Ex. sec. 


.10899 


10918 
10926 
10939 
10952 
10965 
10979 
10992 
11005 
11019 
11032 
11045 
11058 
11072 
11085 
11098 
11112 


* 11125 


.11138 
.11152 
11165 


11178 
.11192 
11205 
L218 
.11282 
11245 
11259 
11272 
11285 
11299 


11812 
11826 
11889 
11353 
11866 
11380 
11893 
.11407 
11420 
11484 


11447 | 


11461 
11474 
11488 
11501 
.11515 
11528 
11542 
11555 
11569 


11583 
11596 
.11610 


11623 


11637 


11651 
11664 


11678 
.11692 
11705 





| 
| 
| 


£12233 
12249 
12266 
12283 
12299 
12316 
12333 
12349 
12366 
. 12883 
12400 


12416 
12483 
12450 
12467 
12484 
- 12501 
12518 
. 12534 
12551 
12568 


12585 
12602 
12619 
12636 
12653 
12670 
12687 
12704 
12721 
- 12788 


£12755 
ART? 
12789 
12807 
12824 
12841 
12858 
12875 
12892 
12910 


12927 
12944 
12961 
12979 
12996 
- 13013 
13031 
13048 
18065 
. 13083 


. 13100 
.13117 
138185 
13152 
13170 
. 13187 
18205 
18222 
_ 18240 
18257 


ODVWOHURWWHO 

















208 TABLE XXIX.—NATURAL VERSED SINES 





28° "  gge 30° 31° 


We lo ef ee ee ee ee 


Vers. |Ex.sec.|| Vers. |Ex.sec.|| Vers. |Ex.sec.|| Vers. |Ex. sec. 








.11705 | .18257 || .12538 | .14335 || .18397 | .15470 || .14283 | .16663 
11719 | .18275 || .12552 | .14354 || .138412 | .15489 || .14298 | .16684 
£11733 | .18292 |} .12566°| .14372 |} .18427 | .15509 |) .14813 | .16704 
11746 | .13310 || .12580 | .14391 || .18441 | .15528 || .14828 | .16725 
.11760 | .18327 || .12595 | .14409 || .13456 | .15548 || .14843 | .16745 
Seed .13345 || .12609 | .14428 || .13470 | .15567 || .14858 | .16766 
.11787 | .13362 || .12623 | .14446 || .18485 | .15587 || .14373 | .16786 
.11801 | .13880 |} .12637 | .14465 |} .18499 | .15606 || .14388 | .16806. 
.11815 | .18398 || .12651 | .14483 || .18514 | .15626 || .14403 | .16827 
£11828 | .138415 |} .12665 | .14502 || .18529 , .15645 || .14418 , .16848 
10 | .11842 | .18483 || .12679 | .14521 || .18543 | .15665 || .14433 | .16868 


11 | .11856 | .18451 |] .12694 | .14589 || .18558 | .15684 || .14449 | .16889 | 11 
12 | .11870 | .13468 || .12708 | .14558 || .18573 | .15704 || .14464 | .16909 | 12 
13 | .11888 | .13486 || .12722 | .14576 || .13587 | .15724 || .14479 | .16930 | 13 
14 | .11897 | .18504 || .12786 | .14595 || .18602 | .15743 || .14494 | .16950 | 14 
15 | .11911 | .18521 || .12750 | .14614 || .18616 | .15763 || .14509 | .16971 | 15 
16 | .11925 | .18539 || .12765 | .14632 || .18631 | .15782 || .14524 | .16992 | 16 

7 | .11988 | .18557 || .12779 | .14651 || .13646 | .15802 || .14539 | .17012 | 17 
18 | .11952 | .138575 || .12798 | .14670 || .13660 | .15822 || .14554 | .17088 | 18 
19 | .11966 | .18593 || .12807 | .14689 |} .138675 | .15841 || .14569 | .17054 | 19 
20 | .11980 | .13610 || .12822 | .14707 || .18690 | .15861 || .14584 | .17075 | 20 


21 | .11994 | .13628 || .12836 | .14726 || .18705 | .15881 || .14599 | .17095 | 21 
22 | .12007 | .18646 || .12850 | .14745 || .18719 | .15901 || .14615 | .17116 | 22 
23 | .12021 | .18664 || .12864 | .14764 |} .187384 | .15920 || .14680 | .171387 | 28 
24 | .12035 | .18682 || .12879 | .14782 || .13749 | .15940 || .14645 | .17158 | 24 
25 | .12049 | .13700 || .12893 | .14801 || .13763 | .15960 || .14660 | .17178 | 25 
26 | .12063 | .18718 || .12907 | .14820 || .13778 | .15980 || .14675 | .17199 | 26 
27 | .12077 | .137385 || .12921 | .14839 || .13793 | .16000 || .14690 | .17220 | 27 
28 | .12091 | .13753 || .12936 | .14858 || .13808 | .16019 || .14706 | .17241 | 28 
29° | .12104 | .187'71 || .12950 | .14877 |) .18822 | .16089 || .14721 | .17262 | 29 
30 | .12118 | .18789 || .12964 | .14896 || .18837 | .16059 || .14786 | .17283 | 30 


31 | .121382 | .18807 || .12979 | .14914 || .138852 | .16079 || .14751 | .17304 | 31 
32 | .12146 | .18825 || .12993 | .14933 || .13867 | .16099 || .14766 | .17825 | 32 
33 | .12160 | .18843 || .18007 | .14952 || .18881 | .16119 || .14782 | .17346 | 33 
34 | .12174 | .18861 |] .18022 | .14971 || .13896 | .16189 || .14797 | .17867 | 34 
35 | .12188 | .18879 || .13036 | .14990 |] .18911 | .16159 || .14812 | .17388 | 35 
36 | .12202 | .13897 || .138051 | .15009 || .13926 | .16179 || .14827 | .17409 | 36 
37 | .12216 | .18916 || .18065 | .15028 || .18941 | .16199 || .14843 | .17430 | 37 
38 | .12830 | .138934 || .18079 | .15047 || .13955 | .16219 || .14858 | .17451 | 38 
39 | .12244 | .18952 || .13094 | .15066 || .13970 | .16239 || .14873 | .17472 | 39 
40 | .12257.| .18970 || .18108 | .15085 || .18985 | .16259 || .14888 | .17493 | 40 


41 | .12271 | .18988 || .18122 | .15105 || .14000 | .16279 |) .14904 | .17514 | 41 
42) .12285 | .14006 || .18187 | .15124 || .14015 | .16299 || .14919 | .175385 | 42 
43 | .12299 | .14024 || .13151 | .15148 || .140380 | .16319 || .14934 | .17556 | 43 
44 | .12313 | .14042 || .138166 | .15162 || .14044 | .16339 || .14949 | .17577 | 44 
45 | .12327 | .14061 |} .18180 | .15181 |} .14059 | .16359.|| .14965 | .17598 | 45 
46 | .12341 | .14079 || .18195 | .15200 || .14074 | .16380 || .14980 | .17620 | 46 
47 | .123855 | .14097 || .138209 | .15219 || .14089 | .16400 |) .14995 | .17641 | 47 
48 | .12369 , .14115 |} .18223 | .15239 || .14104 | .16420 || .15011 | .17662 | 48° 
49 | .12388 | .14184 |} .13238 | .15258 || .14119 | .16440 |) .15026 | .17683 | 49 
50 | .12897 | .14152 || .18252 | .15277 || .14134 | .16460 || .15041 | .17704 | 50 


51 | .12411 | .14170 || .18267 | .15296 |} .14149 | .16481 || .15057 | .17726 | 51 
52} .12425 | .14188 || .18281 | .15315 || .14164 | .16501 || .15072 | .17747 | 52 
53) .12489 | .14207 || .18296 | .15335 || .14179 | .16521 || .15087 | .17768 | 53 
54} £12454 | .14225 || .133810 | .15854 || .14194 | .16541 || .15103 | .17790 | 54 
55 | .12468 | .14243 || .138325 | .15373 |} .14208 | .16562 || .15118 | .17811 | 55 
56 | .12482 | .14262 || .138339 | .15393 |) .14223 | .16582 || .15134 | .17882 | 56 
57 | .12496 | .14280 || .138354 | .15412 || .14238 | .16602 || .15149 | .17854 | 57 
58 | .12510 | .14299 || .133868 | .15431 || .14253 | .16623 || .15164 | .17875 | 58 
59 | .12524 | .14317 || .138383 | .15451 || .14268 | .16643 || .15180 | .17896 | 59 
60 | .12588 | ,14885 |! .13897 | .15470 |! .14283 | ,16663 |! .15195 | .17918 | 60 


a a RS a a SE NE 


| 
| 
| 
| 
| 











WOON wronte | 
=? 
COMBNIATIR WOH OS 



































AND EXTERNAL SECANTS 209 








32° 33° 34° 35° 


~ 
= 


Vers. |Ex.sec.|| Vers. |Ex.sec.|} Vers. |Ex.sec.|| Vers. |Ex. sec. 











15195 | .17918 |} .16133 | .19286 || .17096 | .20622 || .18085 | .22077 | 0 
15211 | .17939 || .16149 | .19259 |} .17113 | .20645 || .18101 | .22102 | 1 
45226 | .17961 || .16165 | .19281 || .17129 | .20669 |} .18118 | .22127 | 2 
15241 | .17982 |} .16181 | ..19304 |} .17145 | .20693 |] .18135 | .22152 ) 3 
15257 | .18004 || .16196 | .19327 || .17161 | .20717 |) .18152 | .22177 | 4 
(15272 | .18025 || .16212 | .19349 || .17178 | .20740 || .18168 | .22202 | 5 
(15288 | .18047 |} .16228 | .19372 |} .17194 | .20764 || .18185 | 22227) 6 
15303 | .18068 |} .16244 | .19394 |, .17210 | .20788 || .18202 | .22252) 7 
“15319 | .18090 || .16260 | .19417 || .17227 | .20812 || .18218 | .22277 | 8 
15334 | .18111 || .16276 | .19440 |) .17243 | .20836 || .18285 | .22302 | 9 
10 | .15350 | .18133 || .16292 | .19463 |) .17259 | .20859 || .18252 | .223827 | 10 


11 | .15365 | .18155 || .16308 | .19485 | .17276 | .20883 |} .18269 | .22352 | 11 
12 | .15381 | .18176 || .16324 | .19508 |) .17292 | .20907 |} .18286 | .22377 | 12 
13 | .15396 | .18198 |] .16340 ; .19531 |) .17308 | .20931 |} .18302 | .22402 | 13 
14 | .15412 | .18220 || .16855 | .19554 |) .17825 | . 20955 || .18879 | .22428 | 14 
15 | .15427 | .18241 || .16371 | .19576 || 17341 | .20979 || .18336 | .22453 | 15 
16 | .15443‘) .18263 || .16387 | .19599 || .173857 | .21003 || .18353 | .22478 | 16 
17 | .15458 | .18285 || .16403 | .19622 || .17374 | .21027 || .18869 | .22503 | 17 
18 | .15474 | .18307 |} .16419 | .19645 || .17390 | .21051 || .183886 | .22528 | 18 
19 | .15489 | .18328 || .16435 | .19668 || .17407 | .21075 || .18403 | .22554 | 19 
20 | .15505 | .18350 || .16451 | .19691 || .17423 | .21099 || .18420 | .22579 | 20 


91 | .15520 | .18372 |} .16467 | .19713 || .1'7439 | .21128 || .18437 | .22604 | 21 
92 | .15536 | .18394 || .16483 |-.19736 || .17456 | .21147 || .18454 | .22629 | 22 
23 | .15552.' .18416 || .16499 | .19759 || .17472 | .21171 || .18470 | .22655 | 23 
94] .15567 | .18437 || .16515 | .19782 || .17489 | .21195 || .18487 | .22680 | 24 
25 | 15583 | .18459 || .16531 | .19805 || .17505 | .21220 || .18504 | .22706 | 25 
96 | .15598 | .18481 || .16547 | .19828 || .17522 | .21244 || .18521 | .22781 | 26 
97 | .15614 | .18503 || .16563 | .19851 || .17538 | .21268 || .18538 | .22756 ) 27 
98 | .15630 | .18525 || .16579 | .19874 || .17554 | .21292 || .18555 | .22782 | 28 
99 | .15645 | .18547 || .16595 | .19897 || .17571 | .21816 || .18572 | .22807 | 29 
30 | .15661 | .18569 || .16611 | .19920 || .17587 | .21841 || .18588 | .22833 | 30 


31 | .15676 | .18591 || .16627 | .19944 || .17604 | .21865 || .18605 | .22858 | 31 
32 | 15692 | .18613 || .16644 | .19967 || .17620 | .21389 || .18622 | .22884 | 32 
33 | .15708 | .18635 || .16660 | .19990 || .17637 | .21414 || .18689 | .22909 | 33 
34 | .15723 | .18657 || .16676 | .20013 || .17653 | .21438 || .18656 , .22935 | 34 
35 | 15739 | .18679 || .16692 | .20036 || .17670 | .21462 || .18673 | .22960 | 35 
36 | .15755 | .18701 || .16708 | .20059 || .17686 | .21487 || .18690 | .22986 | 36 
37 | .15770 | .18723 || .16724 | .20083 || .17703 | .21511 || .18707 | .23012 | 37 
38 | .15786 | .18745 || .16740 | .20106 || .17719 | .21535 || .18724 | .23087 | 38 
39 | .15802 | .18767 || .16756 | .20129 || .17736 | .21560 || .18741 | .238063 | 39 
40 | .15818 | .18790 || .16772 | .20152 || .17752 | .21584 || .18758 | .23089 | 40 


4 | .15833 | .18812 || .16788 | .20176 || .17769 | .21609 |} .18775 | .23114 | 4i 
42 | 15849 | .18834 || .16805 | .20199 || .17786 | .21633 || .18792 | .23140 | 42 
43 | .15865 | .18856 || .16821 | .20222 || .17802 | .21658 || .18809 | .23166 | 43 
44 | .15880 | .18878 || .16837 | .20246 || .17819 | .21682 |; .18826 | .23192 | 44 
45 | .15896 | .18901 || .16853 | .20269 || .17835 | .21707 || .18843 } .28217 | 45 
46 | .15912 | .18923 || .16869 | .20292 || .17852 | .21731 || .18860 | .23243 | 46 
47 | .15928 | .18945 || .16885 | .20816 || .17868 | .21756 || .18877 | .28269 | 47 
48 | .15943 | .18967 || .16902 | .20389 || .17885 | ..21781 || .18894 | .28295 | 48 
49 | .15959 | .18990 || .16918 | .20363 || .17902 | .21805 || .18911 | .28321 | 49 
50 | .15975 | .19012 || .16934 | .20386 || .17918 | .21830 |} .18928 | .23347 | 50 


1 | 15991 | .19084 |) .16950 | .20410 |} .17985 | .21855 |] .18945.} .23373 | 51 
52 | .16006 | .19057 || .16966 | .20433 || .17952 | .21879 || .18962 | .28399 | 
53 | .16022 | .19079 || .16983 | .20457 || .17968 | .21904 |} .18979 | .28424 | 53 
54 | .16038 | .19102 | 16999 | .20480 |! .17985 | .21929 || .18996 | .28450 | 54 
55 | .16054 | .19124"|) .17015 | .20504 || .18001 | .21953 || .19013 | .23476 | 55 
56 | 16070 | .19146 || .17081 | .20527 || .18018 | .21978 || .19080 | .28502 | 56 
57 | 16085 | .19169 || .17047 | .20551 || .18035 | .22003 |} .19047 | .28529 | 57 
58 | 16101 | .19191 || .17064 | .20575 || .18051 | .22028 || .19064 | .23555 | 58 
59 | 16117 | .19214 || .17080 | .20598 || .18068 | .22053 || .19081.| .23581 | 59 
60 | 116133 | .19286 || .17096 | .20622 |! .18085 | .22077 || .19098 | .23607 | 60 
ee ea a a aA aA A Taiko 





DO WIHUCWWHS | 





























Or 
~) 


























910 TABLE XXIX.—NATURAL VERSED SINES 


sais seessnesessesssnssnnssas 


36° 


Vers. |Ex. sec. 


23607 

23683 
28659 
23685 
23711 
23738 
23764 
-23790 
23816 
23843 
23869 


23895 
23922 
23948 
23975 
24001 
«24028 
24054 
24081 
24107 
24134 








0 | .19098 

1 | .19115 

2 | .19133 

3 | .19150 

4} .19167 

B | .19184 

6 | .19201 

7 | .19218 

8 | .19285 

9 | .19252 
10 | .19270 
11 | .19287 
12 | .19804 
13 | .19321 
14 | .19388 
15 | .19356 
16 | .19373 
17 | .19390 
18 | .19407 
19 | .19424 
20 | .19442 
21 | 19459 
22 | 19476 
23 | .19493 
24 | .19511 
25 | .19528 
26 | .19545 
Q7 | .19562 
28 | .19580 
9 | .19597 
30 | .19614 
31} .19632 
32 | .19649 
33 | .19666 
84 | .1y684 
35 | .19701 
36 | .19718 
37 | .19736 
88 | .19753 
39 | .19770 
40 | .19788 
41 | .19805 
42 | .19822 
43 | .19840 
44 | 19857 
45 | .19875 
46 | 19892 
47 | .19909 
48 | 19927 
49 | .19944 
50 | .19962 
51 | .19979 
52 | .19997 
53 | .20014 
54 | .20032 
55 | .20049 
56 | .20066 
57 | .20084 
58 | .20101 
59 | .20119 
60 | 20136 


























37° 

Vers. |Ex. sec, 
.20186 | .25214 
-20154 | .25241 
.20171 | .25269 
.20189 | .25296 
20207 | 253824 
20224 | .25351 
20242 | .25379 
.20259 | 25406 
. 2027’ . 25434 
.20294 | .25462 
.20812 | .25489 
20329 | .25517 
20347 | .25545 
20365 | .25572 
20382 | .25600 
20400 | .25628 
20417 | .25656 
20485 | .256838 
20458 | .25711 
20470 | 25789 
20488 | .25767 
.20506 | .25795 
.20523 | ,25823 
.20541 | .25851 
-20559 | 25879 
2057 .25907 
.20594 | .25985 
20612 | .25963 
20629 | .25991 
20647 | .26019 
20665 | .26047 
20682 | .26075 
20700 | .26104 
.20718 | .26132 
.20786 | .26160 
.20758 | .26188 
20771 | 26216 
.20789 | 26245 
.20807 | .26273 
20824 | .26301 
20842 | .26330 
20860. | .26358 
20878 | .26387 
20895 | .26415 
20913 | .26443 
20931 | .26472 
20949 | .26500 
20967 | 26529 
20985 | .26557 
21002 | .26586 
21020 | .26615 
21038 | .26643 
21056 | .26672 
21074 | .26701 
21092 | .26729 
21109 | .26758 
21127 | .26787 
21145 | .26815 
21163 | .26844 
21181 | .26873 
21199 


- 26902 














38° 

Vers. |Ex. sec. 
21199 | .26902 
21217 | .269381 
21235 | .26960 
.21253 | .26988 
.2127 .27017 
21289 | .27046 
21307 | .27075 
21324 | .27104 
21842 | .27133 
21360 | .27162 
21878 | .27191 
21396 | .27221 
21414 | .27250 
21482 | .27279 
21450 | .27808 
21468 | .27837 
21486 | .27866 
21504 | .27396 
21522 | .27425 
.21540 | .27454 
-21558 | .27483 
. 2157 .275138 
21595 | .27542 
216138 | 2757 

216381 | .27601 
21649 | .27630 
21667 | .27660 
21685 | .27689 
-21703 | .27719 
sR1G21-| {27748 
21789 | RIT 

21757 | .27807 
21775 | ..27887 
21794 | .27867 
21812 | .27896 
21880 | .27926 
21848 | .27956 
21866 | .27985 
21884 | .28015 
21902 | .28045 
21921 | .28075 
21939 | .28105 
21957 | .28184 
21975 | .28164 
219938 | .28194 
22012 | .28224 
22080 | .28254 
22048 | .28284 
.22066 | .28314 
.22084 | .28344 
.22103 | .28874 
22121 | .28404 
22139 | .28484 
22157 | .28464 
22176 | .28495 
22194 | .28525 
22212 | .28555 
22281 | .28585 
.22249 | .28615 
22267 | 28646 
.22285 | .28676 














39° 


Vers. |Ex. séc. 


«22285 
22804 
22922 
22340 
22359 
2207 

22395 
22414 
22432 
22450 
- 22469 
22487 
22506 
22524 
22542 
22561 
22579 
22598 
22616 
22634 
22653 


22671 
22690 
22708 
22020 
222745 
22764 
22782 
22801 
22819 
22838 


22856 
22875 
. 22893 
22912 
22930 
22949 
22967 
22986 
23004 
23023 


23041 
23060 
.23079 
28097 
.23116 
23184 
.23153 
23172 
23190 
238209 


£23228 
23246 
23265 
23283 
23302 
23321 
«23339 
23358 
23377 
23396 





28676 
28706 
28737 
28767 
28797 
28828 
28858 
28889 
28919 
28950 
-28980 


29011 
«29042 
-29072 
-29103 
291383 
-29164 
29195 
29226 
«29256 
«29287 


- .29318 
29349 
29380 
29411 
29442 
29473 
29504 
29535 
29566 
29597 


,29628 
29659 
29690 
29721 
29752 
29784 
.29815 
29846 
29877 
-29909 


.29940 
29971 
30003 
30034 
30066 
30097 
30129 
30160 
30192 
30223 
80255 
30287 
.30318 
30350 
30382 
.BO413 
30445 
30477 
30509 
30541 








OMIDMUIRPWMWEH OS 


10 








AND EXTERNAL SECANTS 


43° 





24529 


| 
| 


40° 

Vers, |Ex. sec. 
.28396 | .80541 
.238414 | .80573 
-23433 | .80605 
.23452 | .30636 
-23470 | .80668 
-23489 | .80700 
.23508 | .30732 
23527 | 380764 
23545 | .80796 
23564 | .80829 
.28583 | .80861 
.23602 | .80893 
. 23620 | .80925 
.238639 | .80957 
.28658 | .80989 
.28677 | .81022 
. 23696. | .81054 
.238714 | ..81086 
.28783 | .381119 
128752 | .<81151 
2377 .381183 
.28790 | .31216 
.23808 | .81248 
.23827 | .81281 
.28846 |- .381313 
.23865 | .81346 
.23884 | .31378 
,209038 | .81411 
-28922 | .31443 
20941 | .381476 
.23959 | .81509 
.23978 | .31541 | 
28997 ) .31574 
.24016 | .81607 
.24085 | .81640 
.24054 | .81672 
.24073 | .81705 
.24092 | .317388 
24111 | sAal771 
.241380 | .381804 
.24149 | ,81837 
.24168 | .381870 
.24187 | .819038 
.24206 | .319386 
. 24225 | .81969 
.24244 | .382002 
. 24262 | 32085 
.24281 | 82068 
.24300 | .82101 
. 24320 | .82134 
, 24389 | .82168 
24358 | .82201 
24877 | .82284 
.24396 | .382267 
24415 | ,32801 
24484 | 323384 
.24453 | .382368 
.24472 | ,82401 
24491 | ,82484 
24510 | ..382468 

.82501 





41° 


Vers. |Ex. sec. 


24548 
24567 
24586 
24605 
24625 
246044 
24663 
24682 
24701 
24720 
24739 
24759 
R477 

24797 
24816 
24835 
24854 
24874 
24893 
24912 


24931 
24950 
24970 
124989 
£25008 
125027 
25047 
£25066 
"25085 
125104 


25124 


25143 
20162 
.250182 
.20201 
25220 
-25240 
20259 
202718 
20297 


25317 
25336 
25356 
25375 
25394 
.20414 
20433 
25452 
20472 
20491 


.20011 
29930 
«20549 
25569 
25588 
29608 
25627 
25647 
25666 


25686 


32501 


82535 
82568 
82602 
32636 
. 32669 
82703 
82737 
82770 
32804 
82838 
82872 
82905 
32939 
382973 
383007 
33041 
83075 
.33109 
383143 
.83177 
83211 
33245 
338279 
.33314 
33348 
33382 
.33416 
83451 
338485 
383519 


383554 
.33588 
383622 
93657 
33691 
33726 
383760 
33795 
33830 
33864 


33899 
383934 
33968 
384003 
34088 
34073 
34108 
84142 
84177 
34212 


84247 
84282 
84317 
84352 
.384387 
84423 
34458 
84493 
34528 




















84563 


42° 
| 


Vers, Ex.sec. 


| 


25686 | .84563 
-25705 | .384599 
25724 | .84634 
.20744 | .84669 
25763 | .34704 
.25783 | .34740 
-25802 | .84775 
20022 | .84811 
.20841 | .384846 
.25861 | .34882 





25880 | 184917 


-25900 | .84953 
-25920 | .84988 
-25989 | .85024 
-25959 | .85060 
25978 | .385095 


.25998 | .85131 


-26017 | .85167 
26037 | .85203 
.26056 | .35238 
26076 | .3527 


.26096 | .35310 
-26115 | .85346 
-26135 | .385382 
.26154 | .35418 
26174 | .85454 


..26194 | .385490 


26213 | .385526 
26283 | .85562 
26253 | .85598 
26272 | .35634 


26292 | .35670 
.26312 | .85707 
26331 | .385743 
-26851 | .85779 - 
-26871 | .85815 
26390 | .385852 
.26410 | .35888 
26480 | .35924 


.26449 | 85961 


-26469 | .85997 


.26489 | ,86054 
.26509 | .86070 
.26528 | ,86107 
.26548 | .86143 
.26568 | 86180 
.26588 | .86217 
26607 | .86253 
26627 | .386290 
26647 | .86327 
.26667 | .386363 


26686 | .36400 
.26706 | 86437 
£26726 | 36474 
.26746 | .386511 
.26766 | .86548 
.26785 | .386585 
.26805 | .86622 
.26825 | .86659 











.26845 | 386696 
.26865 | .36733 





211 


Vers. |Ex. sec. 


Ce 


26865 
26884 
26904 
26924 
26944 
26964 
26984 
27004 
27024 
27043 
27063 


27083 
.271038 
620123 
227148 
27163 
27183 
£27203 
27223 
27243 
27263 


£27283 
27303 
2122 

27343 
27363 
27383 
-27403 
27423 
27443 
21463 
274553 
27508 
220923 
27543 
27563 
27583 
27603 
621623 
27643 
27663 
27683 
227703 
220728 
2270143 
27764 
27784 
27804 
27824 
27844 
27864 
27884 
27905 
27925 
24945 
27965 
27985 
28005 
28026 


28046 © 


28066 


38668 
388705 
| .88744 
88783 
38822 
88860 
88899 
388938 
88977 


89016 


=H Soonccmwwro| 





DD TABLE XXIX.—NATURAL VERSED SINES 





44° 45° 46° 47° 


Vers. |Ex.sec.|| Vers. |Ex.sec.|| Vers. |Ex.sec.|| Vers. |Ex.sec. 


ae | fl | | |  —_ | | | | __ __| __, 


0 | .28066 | .89016 || .29289 | .41421 || .80534 | .43956 || .81800 | .46628 
1 | .28086 | .89055 || .29310 | .41463 || .30555 | .48999 || .81821 | .46674 
2 | .28106 | .89095 || .29330 | .41504 || .80576 | .44042 || .31843 | .46719 
3 | .28127 | .89134 || .29351 | .41545 || .80597 | .44086 || .381864 | .46765 
4] .28147 | .89173 || .29872 | .41586 || .80618 | .44129 || .381885 | .46811 
5 | .28167 | .39212 || .29392 | .41627 || .30639 | .44173 || .81907 | .46857 
6 

7 

8 

9 
10 











-28187 | .89251 || .29413 | .41669 |; .380660 | .44217 || .31928 | .46903 
.28208 | .389291 |} .29433 | .4i1710 || .80681 | .44260 |; .31949 | .46949 
28228 | .39330 || .29454 | .41752 || .80702 | .44304 || .31971 | .46995 ° 
28248 | .389369 || .29475 | .41793 || .80723 | .44347 || .381992 | .47041 
28268 | .389409 || .29495 | .41835 || .80744 | .44391 || .82013 | .47087 


11 | .28289 | .39448 || .29516 | .41876 || .80765 | .44435 || .82085 | .47134 | 11 
12 | .28309 | .89487 || .29537 | .41918 || .380786 | .44479 || .32056 | .47180 | 12 
13 | .28829 | .89527 || .29557 | .41959 || .80807 | .44523 || .382077 | .47226 | 13 
14 | .28850 | .89566 || .29578 | .42001 || .380828 | .44567 || .382099 | .47272 | 14 
15 | .28870 | .389606 || .29599 | .42042 || . , -82120 | .47319 | 15 
16 | .28390 | .89646 || .29619 | .42084 || .380870 | .44654 || .32141 | .47865 | 16 
17 | .28410 | .89685 || .29640 | .42126 |] .380891 | .44698 || .32163 | .47411 | 17 
18 | .28431 | .389725 || .29661 | .42168 || .380912 | .44742 || .82184 | .47458 | 18 
19 | .28451 | .89764 || .29681 | .42210 || .809383 | .44787 || .82205 | .47504 | 19 
20 | .28471 | .89804 || .29702 | .42251 || .80954 | .44831 |] .32227 | .47551 | 20 


21 | .28492 | .39844 || .29723 | .42293 || .80975 | .4487, 32248 | .47598 | 21 
22 | .28512 | .389884 || .29743 | .42335 || .380996 | .44919 || .82270 | .47644 | 22 
23 | .28532 | .39924 || .29764 | .42377 || .381017 | .44963 || .382291 | .47691 | 23 
24 | .28553 | .39963 || .29785 | .42419 || .310388 | .45007 || .82812 | .47788 | 24 
25 | .28573 | .40003 || .29805 | .42461 || .381059 | .45052 || .82334 | .47784 | 25 
26 | .28593 | .40043 || .29826 | .42503 |] .81080 | .45096 || .82355 | .47831 | 26 
27 | .28614 | .40083 || .29847 | .42545 |} .81101 | .45141 || .82377 | .47878 | 27 
28 | .28634 | .40123 || .29868 | .42587 || .81122 | .45185 || .82398 | .47925 | 28 
29 | .28655 | .40163 || .29888 | .42630 || .81143 | .45229 || .82420 | .47972 | 29 
30 | .28675 | .40203 || .29909 | .42672 || .81165 | .45274 |} .82441 | .48019 | 30 


31.| .28695 | .40243 || .29930 | .42714 || .81186 | .45319 || .32462 | .48066 | 31 
32 | .28716 | .40283 || .29951 | .42756 || .81207 | .45363 || .82484 | .48113 | 32 
33 | .287386 | .40324 |) .29971 | .42799 || .81228 | .45408 82505 | .48160 | 33 
34 | .28757 | .40364 || .29992 | .42841 || .381249 | .45452 || .32527 | .48207 | 34 
85 | .28777 | .40404 || .30013 | .42883 || .81270 | .45497 || .82548 | .48254 | 35 
36 | .28797 | .40444 || .80034 | .42926 || .81291 | .45542 || .82570 | .48301 | 36 
37 | .28818 | .40485 || .80054 | .42968 || .81812 | .45587 || .32591 | .48349 | 37 
88 | .28838 | .40525 || .380075 | .43011 || .31334 | .45631 || .82613 | .48396 | 38 
39 | .28859 | .40565 |) .80096 | .43053 || .81855 | .45676 || .82634 | .48443 | 39 
40 | .28879 | .40606 || .80117 | .48096 || .81876 | .45721 || .82656 | .48491 | 40 


41 | .28900 | .40646 || .30138 | .43139 || .81897 | .45766 || .82677 | .48538 | 41 
42 | .28020 | .40687 || .30158 | .43181 ||. .81418 | .45811 || .82699 | .48586 | 42 
43 | .28041 | .40727 || .380179 | .43224 || .81489 | .45856 || .82720 | .48633 | 43 
44 | .28961 | .40768 || .380200 | .43267 || .81461 | .45901 || .82742 | .48681 | 44 
45 | .28981 | .40808 || .380221 | .43310 || .81482 | .45946 || .82763 | .48728 | 45 
46 | .29002 | .40849 || .80242 | .48352 || .31503 | .45992 || .82785 | .48776 | 46 
47 | .29022 | .40890 || .30263 | .43395 |; .81524 | .46037 ;| .82806 | .48824 | 47 
48 | .29043 | .40930 || .80283 | .43438 || .81545 | .46082 |] .82828 , .48871 | 48 
49 | .29063 | .40971 || .30304 | .43481 || .81567 | .46127 || .82849 | .489019 | 49 
50 | .29084 | .41012 || .80325 | .43524 || .81588.| .46173 || .82871 | .48967 | 50 


51 | .29104 | .41053 || .80346 | .43567 || .81609 | .46218 || .82893 | .49015 | 51 
52 | .29125 | .41093 || .380367 | .43610 || .81680 | .46263 || .82914 | .49063 | 52 
53 | .29145 | .41134 || .380388 | .43653 || .81651 | 46309 || .32936 | .49111 | 53 
54 | .29166 | .41175 || .30409 | .43696 || .31673 | .46354 || .82957 | .49159 | 54 
55 |} .29187 | .41216 || .80430 | .43739 || .81694 | .46400 || .82979 | .49207 | 55 
56 | .29207 | .41257 || .80451 | .43783 || .81715 | .46445 || .83001 | .49255 | 56 
57 | .29228 | .41298 || .80471 | .43826 || .81786 | .46491 || .383022 | .49303 | 57 
58 | .29248 | .413889 || .380492 | .43869 || .31758 | .46537 |) .83044 | .49351 | 58 
59 | .29269 | .41380 || .30513 | .48912 || .31779 | .46582 || .33065 | .49399 | 59 
60 | .29289 | .41421 || .30534 | .48956 |! .31800 | .46628 || .33087 | .49448 | 60 


— 
SCOOBVIAUIR WMH OS 





isu) 
S&S 
me 
co 
iS 
ny 
oO 
— 
o 















































AND EXTERNAL SECANTS 


re 
eeaasarenns| 


88087 





Vers. |Ex. sec. 


49448 
83109 | .49496 
83130 | .49544 
83152 | .49593 
83173 | .49641 
.83195 | .49690 
83217 | .49738 
33238 | .49787 
83260 | .49835 
83282 | .49884 
.33303 | .49933 
33325 | .49981 
33347 | .50030 
83368 | .5007 

33390 | .50128 
83412 | .50177 
33434 -| 50226 
83455 | .50275 
33477. | .50324 
33499 | .50373 
33520 | .50422 
83542 | .50471 
83564 | .50521 
83586 | .50570 
83607 | -.50619 
.83629 | .50669 
83651 | .50718 
83673 | .50767 
83694 | .50817 
33716 | .50866 
38738 | .50916 
83760 | .50966 
88782 | .51015 
83803 | .51065 
83825 | .51115 
83847 | .51165 
83869 | .51215 
83891 | .51265 
33912 | .51314 
33984 | .51364 
33956 | .51415 
83978 | .51465 
34000 | .51515 
84022 | .51565 
34044 | 51615 
34065 | .51665 
84087 | .51716 
.34109 | .51766 
84131 | .51817 
34153 | .51867 
84175 | .51918 
84197 | .51968 
34219 | .52019 
84241 | .52069 
34262 | .52120 
84284 | .52171 











49° 
Vers. |Ex. sec. 
.84394 | .52425 
.84416 | .52476 
.04438 | .5252 
.34460 | .52579 
84482 | .52630 
.84504 | .52681 
.84526 | .52732 
.34548 | .52784 
.84570 | .52885 
.84592 | .52886 
.84614 | .52988 
.34636 | .52989 
.84658 | .538041 
.34680 | .53092 
.34702 | .53144 
.34724 | .538196 
.34746 | .53247 
.84768 | .58299 
.384790 | .538851 
.34812 | .53403 
.84834 | .58455 
84856 | .53507 
.384878 | .538559 
.84900 | .53611 
.384923 | .538663 
.84945 | .538715 
-,84967 | .53768 
.84989 | .53820 
-35011 | .538872 
.85033 | .53924 
.85055 | .53977 
.85077 | .54029 
685099 | .54082 
.35122 | .54154 
.35144 | .54187 
.85166 | .54240 
.89188 | .54292 
.85210 | .54845 
.85232 | .54398 
.85254 | .54451 
e277 | .54504 
.85299 | .54557 
.85821 | .54610 
.853843 | .54663 
.35365 | .54716 
.85388 | .54769 
.85410 | .54822 
.85432 | .54876 
.35454 | .54929 
.35476 | .54982 
.85499 | .55036 
.85521 | .55089 
.859438 | .55143 
.385565 | .55196 
680588 | .55250 
35610 | .55303 
$5682 | .55857 
35654 55411 
85677 | .55465 
85699 | .55518 
35721 55572 











.56978 
37000 
.87028 
.87045 
87068 








50‘ 
Vers. |Ex.sec. 
sootel | 755572 

r 385744 | .55626 
.385766 | '.55680 
.85788 | .55734 
.85810 | .55789 
.858383 | .55843 
.80855 | .55897 
00077 | .55951 
.80900 | .56005 
.385922 | .56060 
.35944 | .56114 
.85967 | .56169 
.35989 | .56223 
.36011 | .56278 
.36034 | .56332 
.36056 | .56387 
.36078 | .56442 
.86101 | .56497 
»36123 | -.56551 
.36146 | .56606 
.36168 | .56661 
.36190 | .56716 
.386213 | .56771 
.86235 | .56826 
.86258 | .56881 
.36280 | .56937 
.86302 | .56992 
. 386825 57047 
36347 571038 
.86370 7158 
. 86892 57213 
.36415 57269 
06487 57324 
. 36460 57380 
. 36482 57436 
86504 57491 
.36527 (547 
.386549 57603 
.30572 57659 
86594 57715 
86617 (aval 
.86639 | .5782 
.36662 | .57883 
. 36684 57939 
. 36707 57995 
386729 58051 
. 386752 58108 
86775 58164 
.86797 | .58221 
.86820 | .58277 
. 86842 58333 
.86865 | .58390 
.B6887 | .58447 
.86910 | .58503 
.86932 | .58560 
.86955 | .58617 

5867 











51° 


Vers. |Ex. sec. 


87068 | .58902 
37091 | .58959 
37113 | .59016 
37136 | .59073 
37158 | .59180 
37181 | .59188 
37204 | .59245 
87226 | .59302 
37249 | .59360 
37272 | .59418 
37294 | 59475 
87317 | 59538 
37340 | .59590 
87362 | 59648 
37385 | 59706 
37408 | .59764 
37430 | 59822 
37453 | .59880 
37476 | .59938 
37498 | 59996 
37521 | .60054 
37544 | .60112 
30567 | 60174 
37589 | 60229 
37612 | .60287 
37635 | .60346 
37658 | 60404 
37680 | .60463 
37703 | .60521 
37726 | .60580 
37749 | .60639 
37771 | -.60698 
37794 | .60756 
37817 | .60815 
37840 | .60874 
37862 | 60933 
37885 | .60992 
37908 | .61051 
37931 | .61111 
37954 | 61170 
31976 | .61229 
37999 | .61288 
38022 | .61348 
38045 | .61407 
38068 | .61467 
38091 | .61526 
38113 | .61586 
38136 | .61646 
38159 | 61705 
38182 | .61765 
38205 | .61825 
38228 | .61885 
38251 | .61945 
38274 | .62005 
38296 | .62065 
38319 | .62125 


.62185 
62246 
62306 
62866 
62427 


88342 
38865 
38388 
38411 








Fe a cel el el cell ee oe 
SOON OUF,WMWr Som sonmewno| 


214 


TABLE XXIX.—NATURAL VERSED 


5Q° 


Vers. |Ex. sec. 


fed 
SOOO IOUT WMHS 


88434 
88457 
. 88480 
38503 
. 88526 
88549 
88571 
. 88594 
38617 








.62427 
62487 
62548 
62609 
62669 
62780 
62791 
62852 
62913 
62974 
68035, 


. 63096 
63157 
.63218 
63279 
63841 
. 68402 
68464 
. 63525 
68587 
63648 


.63710 
68772 


65589 
65653 
65717 


65780 


65844. 


.65908 


65972 
660386 
66100 


.66164 

















53° 

Vers. |Ex. sec 
89819 | .66164 
89842 | .66228 
389865 | .66292 
39888 | .66357 
39911 66421 
39985 66486 
39958 | .66550 
89981 | .66615 
40005 | .66679 
40028 | .667. 

40051 | .66809 
40074 | .6687 

40098 | .66938 
40121 | .67003 
40144 67068 
40168 67133 
40191 67199 
40214 7264 
40237 |. .67329 
40261 67394 
40284 67460 
.40807 | .67525 
.40331 67591 
.40354 67656 
.40378 | .67722 
.40401 67788 
.40424 67853 
.40448 67919 
.40471 67985 
.40494 | .68051 
.40518 | .68117 
40541 | .681838 
40565 | .68250 
40588 | .68316 
40611 | .68382 
40635 | .68449 
40658 | .68515 
40682 | .68582 
40705 | .68648 
40728 | .68715 
40752 | .68782 
40775 | .68848 
40799 | 68915 
40822 | .68982 
40846 | .689049 
40869 69116 
40893 | .69183 
40916 69250 
40939 69318 
40963 | .69885 
40986 | .69452 
41010 | .69520 
41033 | .69587 
41057 69655 
41080 | .69723 
41104 69790 




















54° 


Vers. |Ex. sec. 














| 
ae | 
| 


WDIWRO WWE S | 


78215 
18291 
18368 
98445 


SINES 
55° 

Vers. |Ex. sec. 
.42642 | .74345 
.42666 | .74417 
.42690 | .74490 
.42714 | .74562 
.42738 | . 74635 
.42762 | .74708 
42785 | . 74781 
.42809 | .74854 
.42883 | .74927 
.42857 | .75000 
.42881 | .75073 
.42905 | .75146 
.42929 | .75219 
.42953 | .75293 
.42976 | .75366 
.48000 | .75440 
.48024 | .75513 
.438048 | .75587 
.43072 | .75661 
.48096 | .75734 
.48120 | .75808 
.48144 | .75882 
43168 | .75956 
-43192 | .76081 
.48216 | .76105 
.43240 | .76179 
.48264 | .76253 
.48287 | .76328 
.43311 | .76402 
438385 | .'76477 
43859 | .76552 
438383 | .76626 
.438407 | .76701 
.43481 | 27677 
.43455 | .76851 
4347 - 76926 
.485038 | .77001 
-48527 | .77077 
43551 | .77152 
AB51D 1 (R27. 
43599 | .77803 
43623 | .77378 
43647 | 77454 
43671 | 77580 
43695 77606 
43720 | .7'7681 
43744 TST 
43768 | .778383 
43792 nt 
43816 | .77986 
438840 | .78062 
43864 | .78138 





10 


AND EXTERNAL SECANTS 215 


pe ey er se ee ee ee 
56° 57° 58° 59° 
yg | sf | | | 


Vers. |Ex.sec.|| Vers. |Ex.sec.|| Vers. |Ex.sec.]| Vers. Ex. sec. 











44081 | .78829 || .45536 | .83608 || .47008 | .88708 || .48496 | .94160 
44105 | .78906 || .45560 | .838690 || .47033 | .88796 || .48521 | .94254 
44129 | .78984 || .45585 | .83773 || .47057 | .88884 |) .48546 | .94349 
44153 | .79061 |; .45609 | .83855 || .47082 | .88972 || .48571 | .94443 
: 79138 || 45634 | .83938 || .47107 | .89060 || . 94537 
44201 | .79216 || .45658 | .84020 || .47181 | .89148 || .48621 | .94632 
44225 | .79293 || .45683 | .84103 || .47156 | .892387 || . 94726 
.44250 | .79371 || .45707 | .84186 || .47181 | .89825 || .48671 | .94821 
44274 | .79449 || .45731 | .84269 || .47206 | .89414 || .48696 | .94916 
44298 | .79527 || .45756 | .84352 || .47280 | .89503 || .48721 | .95011 
10 | .44822 | .79604 || .45780 | .84435 || .47255 | .89591 || .48746 | .95106 


11 | .44346 | .79682 || .45805 | .84518 || .47280 | .89680 || .487'71 | .95201 | 11 
42 | .44370 | .79761 || .45829 | .84601 || .47304 | .89769 || .48796 | .95296 | 12 
13 | .44395 | .79839 || .45854 | .84685 || .47329 | .89858 ||..48821 | .95892 | 138 
14 | .44419 | .79917 || .45878 | .84768 || .47354 | .89948 || .48846 | .95487 | 14 
$5 | .44443 | .79995 || .45903 | .84852 || .47379 | .90087 || .48871 | .95583 | 15 
16 | .44467.} .80074 || .45927 | .84985 || .47403 | .90126 || .48896 | .95678 | 16 
17 | .44491 | .80152 || .45951 | .85019 || .47428 | .90216 || .48921 | .95774 | 17 
18 | .44516 | .80231 || .45976 | .85103 || .47453 | .90305 || .48946 | .95870 | 18 
19 | .44540 | .80309 || .46000 | .85187 || .47478 | .90395 || .48971 | .95966 | 19 
20 | .44564 | .80388 || .46025 | .8527 .47502 | .90485 || .48996 | .96062 | 20 


91 | .44588 | .80467 || .46049 | .85855 || .47527 | .90575 || .49021 | .96158 | 21 
92 | .44612 | .80546 ||. .46074 | .85439 || .47552 | .90665 || .49046 | .96255 | 22 
93 | .44637 | .80625 || .46098 | .85523 || .47577 | .90755 || .49071 | .96351 | 23 
24 | .44661 | .80704 |} .46123 | .85608 || .47601 | .90845 |) .49096 | .96448 | 24 
95 | .44685 | .80783 || .46147 | .85692 || .47626 | .90935 || .49121 | .96544 | 25 
26 | .44709 | .80862 || .46172 | .8577 47651 | .91026 || .49146 | .96641 | 26 
97 | .44734 | .80942 || .46196 | .85861 || .47676 | .91116 || .49171 | .96738 | 27 
98 | .44758 | .81021 || .46221 | .85946 || .47701 | .91207 || .49196 | .96835 | 28 
99 | .44782 | .81101 || .46246 | .$6081 || .47725 | .91297 || .49221 | .96932 | 29 
30 | .44806 | .81180 || .46270 | .86116 || .47750 | .91388 || .49246 | .97029 | 30 


81 | .44831 | .81260 || .46295 | .86201 || .47775 | .9147 49271 | .97127 | 31 
82 | .44855.} .81340 || .46319 | .86286 |! .47800 | .9157 .49296 | .97224 | 82 
33 | .44879 | .81419 || .46344 | .863871 |) .47825 | .91661 || .49321 | .97322 | 33 
34 | .44903 | .81499 || .46368 | .86457 || .47849 | .91752 || .49346 | .97420 | 34 
35 | .44928 | .81579 || .46393 | .86542 || .47874 | .91844 |) .49872 | .97517 | 35 
36 | .44952 | .81659 || .46417 | .86627 || .47899 | .91935 || .49397 | .97615 | 36 
- 87 | .44976 | .81740 || .46442 | .86718 || .47924 | .92027 || .49422 | .97713 | 37 
38 | .45001 | .81820 || .46466 | .86799 || .47949 | .92118 || .49447 | .97811 | 38 
39 | .45025 | .81900 || .46491 | .86885. || .47974 | .92210 || .49472 | .97910 | 39 
40 | .45049 | .81981 || .46516 | .86970 || .47998 |. .923802 || .49497 | .98008 | 40 


41 | .45073 | .82061 || .46540 | .87056 || .48023 | .92394 || .49522 | .98107 | 41 
42 | .45098 | .82142 |} .46565 | .87142 || .48048 | .92486 || .49547 | .98205 | 42 
43 | .45122 | .82222 || .46589 | .87229 || .48078 | .9257 4957. .98304 | 48 
44 | .45146 | .82303 || .46614 | .87315 || .48098 | .92670 || .49597 | .98403 | 44 
45 | .45171 | .82384 || .46639 | .87401 || .48128 | .92762 || .49623 | .98502 | 45 
46 | .45195 | .82465 || .46663 | .87488 || .48148 | .92855 || .49648 | .98601 | 46 
47 | .45219 | .82546 || .46688 | .87574 || .48172 | .92947 || .49673 | .98700 | 47 
48 | .45244 | .82627 || .46712 | .87661 || .48197 | .93040 || .49698 | .98799 | 48 
49 | .45268 | .82709 || .46737 | .87748 || .48222 | .93133 || .49728 | .98899 | 49 
50 | .45292 | .82790 || .46762 | .87834 || .48247 | .93226 || .49748 | .98998 | 50 


51 | .45317 | .82871 || .46786 | .87921 || .48272 | .93319 || .49773 | .99098 | 51 
52 | .45341 | .82953 || .46811 | .88008 || .48207 | .938412 || .49799 | .99198 | 52 
53 | .45365 | .83034 || .46836 | .88095 || .48322 | .93505 || .49824 | .99298 | 53 
54 | .45390 | .83116 || .46860 | .88183 || .48347 | .93598 || .49849 | .99398 | 54 
55 | .45414 | .83198 ||, .46885 | .88270 || .48872 | .93692 || .49874 | .99498 | 55 
56 | .45489 | .83280 || .46909 | .88357 || .48396 | .93785 || .49899 | .99598 56 
57 | .45463 | .83362 || .46934 | .88445 || .48421 | .93879 || .49924 | .99698 | 57 
58 | .45487 | .83444 |) .46959 | .88532 || .48446 | .9397 .49950 | .99799 | 58 
39 | .45512 | .88526 || .46983 | .88620 || .48471 | .94066 || .49975 | .99899 | 59 
60 | .45536 | .83608 || .47008 | .88708 || .48496 | .94160 |] .50000 '1.00000 | 60 


TT 


is 
@ 
or 
oO 
oS 


A 
_— 
~ 
~ 

& 

oo 

Ris 

for} 


= 
COON WOH OS 






















































































| 
| 
| 
| 
| 


216 TABLE XXIX.—NATURAL VERSED SINES 
eS a A One Ae ey 



















































































f 


60° 61° 62° 63° 

s i) 

Vers. | Ex .se Vers. | Ex. sec.|| Vers, |Ex.sec. || Vers. | Ex. sec.! 
0; .50000 | 1.00000 .51519 | 1.06267 53053 | 1.13005 54601 | 1.20269 | 0 
1} .50025 | 1.00101 .51544 | 1.0637 538079 | 1.13122 .54627 | 1.20895 | 1 
2] .50050 | 1.00202 .51570 | 1.06483 53104 | 1.138239 .546538 | 1.20521 | 2 
8| .50076 | 1.00303 .51595 | 1.06592 531380 | 1.13356 .54679 ' 1.20647 | 3 
4; .50101 | 1.00404 .51621 | 1.06701 538156 | 1.13473 04705 | 1.20778 | 4 
5| .50126 | 1.00505 -51646 | 1.06809 53181 | 1.13590 54731 | 1.20900 | 5 
6| .50151 | 1.00607 .51672 | 1.06918 53207 | 1.13707 .04757 | 1.21026 | 6 
7! .50176 | 1.00708 || *.51697 | 1.07027 53288 | 1.18825 -5A782 | 21,211 58i 7 
8| .50202 | 1.00810 .51728 | 1.07137 58258 | 1.13942 .54808 | 1.21280 | 8 
9| .50227 | 1.00912 .51748 | 1.07246 538284 | 1.14060 .54834 | 1.21407 | 9 
10} .50252 | 1.01014 .51774 | 1.07356 .538310 | 1.14178 .54860 | 1.21535 |10 
11; .50277 | 1.01116 || .51799 | 1.07465 || .53836 | 1.14296 || .54886 | 1.21662 |11 
12) .50803 | 1.01218 .51825 | 1.07575 .53861 | 1.14414 .64912 | 1.21790 |12 
13] .50328 | 1.01320 .51850 | 1.07685 5383887 | 1.14533 .54988 | 1.21918 |13 
14) .50353 | 1.01422 .51876 | 1.07795 534138 | 1.14651 .54964 | 1.22045 |14 
15| .503878 | 1.01525 .51901 | 1.07905 53489 | 1.14770 .54990 | 1.22174 |15 
16| .50404 | 1.01628 .51927 | 1.08015 53464 | 1.14889 .55016 | 1.22302 |16 
17) .50429 | 1.01730 .51952 | 1.08126 53490 | 1.15008 .05042 | 1.22430 |17 
18| .50454 | 1.01833 .51978 | 1.08236 53516 | 1.15127 .55068 | 1.22559 |18 
19| .50479 | 1.01936 .52003 | 1.08347 53542 | 1.15246 .55094 | 1.22688 119 
20) .50505 | 1.02039 .52029 | 1.08458 58567 | 1.15366 .55120 | 1.22817 |20 
21| .50530 | 1.02148 .52054 | 1.08569 535938 | 1.15485 -55146 | 1.22946 | 21 
22| .50555 | 1.02246 .52080 | 1.08680 53619 | 1.15605 .55172 | 1.23075 |22 
23| .50581 | 1.02849 .52105 | 1.08791 53645 | 1.15725 .55198 | 1.23205 | 23 
24] .50606 | 1.02453 521381 | 1.08903 .538670 | 1.15845 55224 | 1.23334 | 24 
25| .506381 | 1.02557 .52156 | 1.09014 .5869 1.15965 -5d250 | 1.28464 125 
26| .50656 | 1.02661 .52182 | 1.09126 .58722 | 1.16085 .55276 | 1.28594 | 26 
27) .50682 | 1.02765 .52207 | 1.09238 53748 | 1.16206 FOOGUe inde ces ae 
28] .50707 | 1.02869 . 52283 | 1.09350 538774 | 1.16326 .55828 | 1.23855 | 28 | 
29| .50732 | 1.029738 52259 | 1.09462 53799 | 1.16447 .55354 | 1.23985 | 
80! .50758 | 1.08077 52284 | 1.09574 538825 | 1.16568 .553880 | 1.24116 | 30 
81] .50783 | 1.03182 || .52310 | 1.09686 53851 | 1.16689 || .55406 | 1.24247 |31 
82| .50808 | 1.03286 .523385 | 1.09799 538877 | 1.16810 .0d432 | 1.24878 
383] .50834 | 1.038391 .523861 | 1.09911 53903 | 1.16932 .55458 | 1.24509 
34] .50859 | 1.03496 || .52386 | 1.10024 538928 | 1.17053 || .55484 | 1.24640 | 34 
35] .50884 | 1.03601 .52412 | 1.10137 53954 | 1.17175 .00510' | 1.24772 135 
36/ .50910 | 1.03706 .524388 | 1.10250 53980 | 1.17297 .55536 | 1.24903 | 36 © 
87| .50985 | 1.08811 .52463 | 1.10363 54006 | 1.17419 .55563 | 1.25085 
88] .50960 | 1.038916 .52489 | 1.10477 54082 | 1.17541 .05589 | 1 25167 |38 
389} .50986 | 1.04022 .52514 | 1.10590 54058 | 1.17663 .55615 | 1.25300 |39 
40} .51011 | 1.04128 || .52540 | 1.10704 54088 | 1.17786 || .55641 | 1.25482 140 
41} .51086 | 1.04288 || .52566 | 1.10817 54109 | 1.17909 || .55667 | 1.25565 | 41 
42) .51062 | 1.04839 -52591 | 1.10931 541385 | 1.18031 .50693 | 1.25697 | 42 
43] .51087 | 1.04445 .52617 | 1.11045 54161 | 1.18154 .55719 | 1.25880 | 43 
44] .51118 | 1.04551 .52642 | 1.11159 54187 | 1.18277 .55745 | 1.25963 
45] .51188 | 1.04658 .52668 | 1.11274 54218 | 1.18401 -bo771 | 1.26097 | 45% 
46) .51163 | 1.04764 .52694 | 1.11388 54238 | 1.18524 .55797 | 1.26280 | 46 
7) .51189 | 1.0487 .52719 | 1.11503 54264 | 1.18648 .55823 | 1.26364 | 47 
48} .51214 | 1.04977 eto in Ll dGii 54290 | 1.18772 .55849 | 1.26498 
49} .51289 | 1.05084 POO Le leelligese 54816 | 1.18895 .55876 | 1.26682 | 49) 
50] .51265 | 1.05191 .52796 |. 1.11847 543842 | 1.19019 .55902 | 1.26766 | 50 
51| .54290 | 1.05298 || .52822 | 1.11963 || .54368 | 1.19144 || .55928 | 1.26900 |51 
52) .51316 | 1.05405 .52848 | 1.12078 -b4894 | 1.19268 .55954 |.1.270385 152 
53| .51841 | 1.05512 .5287 1.12193 .54420 | 1.19893 .55980 | 1.27169 |53 
54| .51366 | 1.05619 .52899 | 1.12309 .54446 | 1.19517 .56006 | 1.273804 |54 
55) .51892 | 1.05727 .52924 | 1.12425 .54471 | 1.19642 .56082 | 1.27439 | 55 
56| .51417 | 1.05835 .52950. | 1.12540 .54497 | 1.19767 .56058 | 1.27574 |56 
57| .514438 | 1.05942 .52976 | 1.12657 .545238 | 1.19892 .56084 | 1.27710 |57 
58| .51468 | 1.06050 .58001 | 1.1277 .54549 | 1.20018 .56111 | 1.27845 |58 
59| .51494 | 1.06158 .538027 | 1.12889 .04575 | 1.20148 .561387 | 1.27981 |59 
60! .51519 | 1.06267 .58053 ' 1.13005 .54601 | 1.20269 .56163 | 1.28117 160 
SS NS pen na none 


ae ee 


AND EXTERNAL SECANTS 7) 







































































64° 65° - 66° 67° 
Oe et ee 
Vers. | Ex. sec.|| Vers. | Ex. sec.!] Vers. | Ex. sec. Vers. | Ex. sec. 
0} .56163 | 1 eas \| 57738 | 1.36620 || -59326 | 1. 45859 .60927 | 1.559380 | 0 
1} .56189 | 1.28253 || .57765 | 1.386768 || .593853 | 1.46020 || .60954 | 1.56106 | 1 
2) 56215 | 1 a0 || .o7791 | 1.36916 || .59379 | 1.46181 || .60980 , 1.56282 2 
8] .56241 | 1.28526 || .57817 | 1.37064 || .59406 | 1.46342 || .61007 | 1.56458 | 3 
4) .56267 | 1.28663 || .57844 | 1.37212 || .59433 | 1.46504 || .61034 | 1.56634 | 4 
5| .56294 | 1.28800 .5787 1.37361 || .59459 | 1.46665 || .61061 | 1.56811 | 5 
6) .56320 | 1.28987 || .57896 | 1.37509 || .59486 | 1.46827 || .61088 | 1.56988 | 6 
7| 56346 | 1.29074 |} .57923 | 1.87658 || .59512 | 1.46989 || .61114 | 1.57165 | 7 
8| .563872 | 1.29211 .57949 | 1.37808 || .59539 | 1.47152 || .61141 | 1.57342 | 8 
9| .56398 | 1.29349 || .57976 | 1.387957 || .59566 | 1.47314 || .61168 | 1.57520 | 9 
10) .56425 | 1.29487 || .58002 | 1.38107 || .59592 | 1.47477 || .61195 | 1.57698 | 10 
11} .56451 | 1.29625 || .58028 | 1.38256 || .59619 | 1.47640 || .61222 | 1.57876 | 11 
12} .5647 1° 29763 .58055 | 1.38406 .59645 | 1.47804 |! .61248 | 1.58054 |12 
13) .56503 | 1.29901 || .58081 | 1.38556 || .59672 | 1.47967 || .61275 | 1.58233 | 13 
14} .56529 | 1.30040 || .58108 | 1.38707 || .59699 | 1.48131 .61302 | 1.58412 | 14 
15; .56555 | 1.30179 .58184 | 1.38857 59725 | 1.48295 | .61329 | 1.58591 |15 
16| .56582 | 1.30318 .58160 | 1.39008 .59752 | 1.48459 .61356 | 1.58771 |16 
7) .56608 | 1.30457 .58187 | 1.39159 25977 1.48624 || .61883 | 1.58950 17 
18| .56634 | 1.30596 || .58213 | 1.89311 .59805 | 1.48789 | .61409 1.59180 |18 
19| .56660 | 1.30735 || .58240 | 1.39462 .59832 | 1.48954 || .61486 | 1.59311 |19 
20) .56687 | 1.80875 || .58266 | 1.39614 || .59859 | 1.49119 || .61463 | 1.59491 | 20 
21| .56713 | 1.31015. || .58293 | 1.39766 || .59885 | 1.49284 || .61490 | 1.59672 |21 
22) .56739. | 1.31155 .58319 | 1.39918 .59912 | 1.49450 .61517 | 1.59853 | 22 
293} .56765 | 1.31295 || .58345 | 1.40070 || .59938 | 1.49616 || .61544 | 1.60035 |23 
24) .56791 | 1.31486 -Q8872 | 1.40222 .59965 | 1.49782 .6157 1.60217 | 24 
25| .56818 | 1.31576 .58898 | 1.403875 .59992 | 1.49948 .61597 | 1.60399 | 25 
26| .56944 | 1.31717 || .58425 | 1.40528 |} .60018 | 1.50115 || .61624 | 1.60581 | 26 
v| 56870 | 1.31858 || .58451 | 1.40681 .60045 | 1.50282 || .61651 | 1.60763 |27 
98! .56896 | 1.31999 || .58478 | 1.40835 || .60072 | 1.50449 || .61678 | 1.60946 | 28 
29| .56923 | 1.32140 || .58504 | 1.40988 |} .60098 | 1.50617 || .61705 | 1.61129 | 29 
80| .56949 | 1.82282 || .58531 | 1.41142 ||. .60125 | 1.50784 || .61782 | 1.61313 | 30 
31| .56975 | 1.82424 || .58557 | 1.41296 || .60152 | 1.50952 || .61759 | 1.61496 |31 
32! .57001 | 1.82566 || .58584 | 1.41450 || .60178 | 1.51120 || .61785 | 1.61680 |82 
33] .57028 | 1.32708 || .58610 | 1.41605 || .60205 | 1.51289 || .61812 | 1.61864 | 33 
34! .57054 | 1.82850 || .58637 | 1.41760 || .60232 | 1.51457 || .61889 | 1.62049 | 34 
35| .57080 | 1.32993 || .58663 | 1.41914 || .60259 | 1.51626 || .61866 | 1.62234 | 35 
86| .57106 |. 1.33185 || .58690 | 1.4207 .60285 | 1.51795 || .61893 | 1.62419 |36 
7! 57133 | 1.38278 || .58716 | 1.42225 || .60312 | 1.51965 || .61920 | 1.62604 |37 
38] .57159 | 1.33422 || .58743 | 1.42380 || .60339 | 1.52134 || .61947 | 1.62790 |38 
39| .57185 | 1.33565 || .58769 | 1.42536 || .60865 | 1.52304 || .61974 | 1.62976 |39 
40| .57212 | 1.33708 || .58796 | 1.42692 || .60892 | 1.52474 || .62001 | 1.63162 | 40 
41| .57238 | 1.33852 || .58822 | 1.42848 || .60419 | 1.52645 || .62027 | 1.63348 | 41 
42} .57264 | 1.33996 .58849 | 1.438005 .60445 | 1.52815 || .62054 | 1.63535 | 42 
43| .57291 | 1.84140 .58875 | 1.43162 .60472 ot .62081 | 1.63722 |43 
44| .57317 | 1.34284 || .58902 | 1.43318 || .60499 | 1.53157 || .62108 | 1.63909 | 44 
45| 57343 | 1.34429 || .58928 | 1.43476 || .60526 153329 || ,62135 | 1.64097 |45 
46| .57369 | 1.34573 || .58955 | 1.43633 .60552 | 1.53500 | .62162 | 1.64285 | 46 
7| .57396 | 1.34718 || .58981 | 1.43790 || .60579 | 1.53672 || .62189 | 1.64478 |47 
48| .57422 | 1.34863 |} .59008 | 1.43948 || .60606 | 1.53845 || .62216 | 1.64662 | 48 
49| 57448 | 1.35009 |} .59034 | 1.44106 || .60633 | 1.54017 || .62243 | 1.64851 | 49 
50| .57475 | 1.85154 || .59061 | 1.44264 || .60659 | 1.54190 || .6227 1.65040 | 50 
51| .57501 | 1.35300 |} .59087 | 1.44423 || .60686 | 1.54363 || .62297 | 1.65229 |51 
52 7527 | 1.35446 .59114 | 1.44582 60713 | 1.54536 ,62324 | 1.65419 | 52 
53| .57554 | 1.35592 || .59140 | 1.44741 || .60740 | 1.54709 || .62351 | 1.65609 |53 
54| .57580 | 1.35738 || .59167 | 1.44900 || .60766 | 1.54883 || .62378 | 1.65799 | 54 
55| .57606 | 1.35885 |4..59194 | 1.45059 || .60793 | 1.55057 .62405 | 1.65989 | 55 
56) .57633 | 1.36031 59220 | 1.45219 || .60820 | 1.55231 || .62481 | 1.66180 |56 
57| .57659 | 1.386178 || .59247 | 1.4587: 60847 | 1.55405 || .62458 | 1.66371 |57 
58| .57685 | 1.36825 |! .59273 | 1.45539 || .60873 | 1.55580 |) .62485 | 1.66563 | 58 
59| .57712 | 1.86473 | .59300 | 1.45699 .60900 | 1.55755 62512 | 1.66755 |59 
60! .57738 | 1.36620 |] .59326 | 1.45859 || .60927 | 1.55930 || .62539 | 1.66947 | 60 








218 











68° 
Vers. | Ex, sec. 
62539 | 1.66947 | 
. 62566 | 1.67139 
.62593 | 1.67332 
62620 | 1.67525 
.62647 | 1.67718 
62674 | 1.67911 
.62701 | 1.68105 
.62728 | 1.68299 
.62755 | 1.68494 
.62782 | 1.68689 
62809 | 1.68884 
2836 | 1.6907 
62863 | 1.69275 
62890 | 1.69471 
62917 | 1.69667 
62944 | 1.69864 
.62971 | 1.70061 
.62998 | 1.70258 
.63025 | 1.70455 
.638052 | 1.70653 
63079 | 1.70851 
63106 | 1.71050 
631383 | 1.'71249 
63161 | 1.71448 
63188 | 1.71647 
68215 | 1.71847 
63242 | 1.72047 
63269 | 1.72247 
63296 | 1.72448 
63323 | 1.72649 
63350 | 1.72850 
638377.| 1.78052 
63404 | 1.738254 
63431 | 1.73456 
63458 | 1.73659 
63485 | 1.73862 
63512 | 1.74065 
63539 | 1.74269 
63566 | 1.74473 
63594 | 1.74677 
63621 | 1.74881 
.63648 | 1.75086 
.68675 | 1.75292 
.63702 | 1.75497 
68729 | 1.75703 
63756 | 1.75909 
63783 | 1.76116 
63810 | 1.76323 
63888 | 1.76530 
63865 | 1.76737 
63892 | 1.76945 
63919 | 1.77154 
63946 | 1.77362 
63973 | 1.77571 
64000 | 1 77780 
64027 | 1.77990 
64055 | 1.78200 
64082 | 1.78410 
64109 | 1.78621 | 
64136 | 1.78832 
64163 | 1.79048 























69° 
Vers. | Ex. sec. 
.64163 | 1.79048 
64190 | 1.79254 
.64218 | 1.'79466 
.64245 | 1.79679 
64272 | 1.79891 
.64299 } 1.80104 
.64826 | 1.80318 
.64853 | 1.80531 
.64381 | 1.80746 
.64408 | 1.80960 
.64435 | 1.81175 
.64462 | 1.81390 
.64489 | 1.81605 
.64517 | 1.81821 
.64544 | 1.82037 
.64571 | 1.82254 |) 
.64598 | 1.82471 
.64625 | 1.82688 
.64653 | 1.82906 
.64680 | 1.83124 
.64707 | 1.83342 
.64734 | 1.83561 
.64761 | 1.838780 
.64789 | 1.838999 
.64816 | 1.84219 
.64843 | 1.84439 
.64870 | 1.84659 
.64898 | 1.84880 
.64925 ; 1.85102 
64952 | 1.85823 
.64979 | 1.85545 
.65007 | 1.85767 
.65084 | 1.85990 
.65061 | 1.86213 
.65088 | 1.86437 
65116 | 1.86661 
.65148 | 1.86885 
.65170 | 1.87109 
.65197 | 1.87334 
.65225 | 1.87560 
.65252 | 1.87785 
65279 | 1.88011 
.65306 | 1.88238 
.65334 | 1.88465 
.65361 | 1.88692 
.65888 | 1.88920 
.65416 | 1.89148 
.65443 | 1.89376 
.65470 | 1.89605 
.65497 | 1.89834 
.65525 | 1.90063 
-65552, | 1.90293 
.65579 | 1.90524 
.65607 | 1.90754 
.65634 | 1.90986 
.65661 | 1.91217 
.65689 | 1.91449 
.65716 | 1.91681 
.65743 | 1.91914 
.65771 | 1.92147 
65798 | 1.92380 

















TABLE XXIX.—NATURAL VERSED 


70° 








2.02057 


2.02308 
2.02559 
2.02810 
2.03062 
2.03315 
2.03568 
2.03821 
2.04075 
2.04329 
2.04584 


2.04839 
2.05094 
2.05350 
2.05607 
2.05864 
2.06121 
2.06379 
2.06637 
2.06896 
2.07155 




















SINES 


ed 


Way 


Vers. | Ex. sec. 





67443 
67471 
67498 
67526 
67553 
67581 
-67608 
67636 
. 67663 
67691 
67718 
67746 
BTT73 
67801 
67829 
67856 
67884 
67911 
67939 
67966 











2.07155 
2.07415 
2.07675 
2.07936 
2.08197 
2.08459 
2.08721 
2.08983 
2.09246 
2.09510 
2.09774 


2.10038 
2.10303 
2.10568 
2.10834 
2.11101 
2.11867 
2.11635 
2.11903 
2.12171 
2.12440 


2.12709 
2.12979 
2.18249 
2.138520 
2.18791 
2.14063. 
2.14835 
2.14608 
2.14881 
2.15155 


2.15429 
2.15704 
2.15979 
2.16255 
2.16531 
2.16808 
2.17085 
2.17363 
2.17641 
2.17920 


2.18199 
2.18479 
2.18759 
2.19040 
2.19R22 
2 1964 
2.19886 
2 20169 
2.20453 
2.20737 


2.21021 
2.21806 
2.21592 
2.21878 
2.22165 
2.22452 
2.22740 
2. 28028 
2.28317 
2.23607 








= 
SweMrIoURmwoHo | 











72° 


Ex, sec. 


2.23897 
2.24187 
2.24478 
2.24770 
2.25062 
2.25855 
2.25648 
2.25942 
2 26237 
2.26531 


2. 26827 
2.27123 
2.27420 
2.27717 
2.28015 
2.28313 
3.28612 
2.28912 
2.29212 
2.29512 


2.29814 
2.30115 
2.30418 
2.20721 
2.31024 
2.31328 
2.31633 
2.31939 
2.382244 
2.82551 


2.32858 
2.23166 
2.33474 
2.33788 
2.34092 
2.34403 
2.34713 
2.35025 
2.85336 
2.35649 


2.85962 
2.86276 
2.86590 
2.36905 
2.37221 
2.375387 
2.87854 
2.38171 
2.38489 
2.38808 


2.39128 
2.30448 
| 2.39768 
2.40089 
2.40411 
2.40734 
2.41057 
2.41381 
2.41705 
2.42030 








2.23607 











AND EXTERNAL SECANTS 


A ar i ae 


Vers. 


10763 
T0791 
70818 
70846 
70874 
70902 
- 70930 
70958 
«T0985 
71013 
-71041 
- 71069 
71097 
«(1125 
71153 
71180 
- 71208 
71236 
- 71264 
71292 
71820 
71348 
1375 
71403 
71481 
71459 
71487 
71515 
71548 
T1571 


«71598 


71626 
71654 
T1682 
71710 
71738 
71766 
T1794 
«71822 
71850 


T1877 . 


71905 
. 71933 
71961 
71989 
(2017 
72045 
12078 
72101 
02129 
02157 
72185 
(2213 
(2241 
(2269 
9 f2296 
T2824 
(2852 
72380 


. 72408 


72436 





73° 


Ex. sec. 


2.42356 
2.42683 
2.48010 
2.43337 
2.48666 
2.48995 
2.44824 
2.44655 
2.44986 
2.45317 
2.45650 
2.45983 
2.46316 
2.46651 
2.46986 
2.47321 
2.47658 
2.47995 
2.48333 
2.48671 


2.49010 
2.49350 
2.49691 
2.50082 
2.50874 
2.50716 
2.51060 
2.51404 
2.51748 
2.52094 


2.52440 
2.52787 
2.531384 
2.58482 
2.53831 
2.54181 
2.54531 
2.54883 
2.55285 
2.55587 
2.55940 
2.56294 
2.56649 
2.57005 
2.57361 
2.57718 
2.58076 
2.58434 
2.58794 
2.59154 


2.59514 
2.59876 





2.60601 
2.60965 
2.61380 
2.61695 
2.62061 
2.62428 
2.62796 





2.42030 








2.60238. |) 


74° 


Vers. 


72486 


12464 
72492 
72520 
112548 
72576 
72604 
12632 
72660 
12688 
72716 
12744 
27 


«72800 
12828 
«72856 
«72884 
(2912 
- (2940 
- 72968 
- 72996 


73024 
7B052 
-73080 
73108 
.73136 
-73164 
73192 
73220 
T3248 
73276 
73304 
73332 
73360 
73388 
“73416 
73444 
73472 
-73500 
{3529 
{3557 
73585 
73613 
"3641 
73669 
“73697 
TB725 
M3753 
73781 
73809 
73837 


73865 
73898 
73921 
73950 
73978 
74006 
74034 
74062 
-74090 
74118 





Ex. sec. 


2.62796 


2.68164 
2.63533 
2.63903 
2.64274 
2.64645 
2.65018 
2.65891 
2.65765 
2.66140 
2.66515 


2.66892 


Ww 
=? 
> 
co 
me 
or 


~> 
Tey 
© 
@ 
nS 


2.82633 
2.83045 
2.83457 
2.83871 
2.84285 
2.84700 
2.85116 
2.85533 
2.85951 
2.86370 











75° 


Vers. 


T4118 


75272 
753800 
75828 
. 75856 
75885 
15418 
05441 
75469 
15497 
. 75526 


Ex. sec. 


2.86370 
2.86790 
2.87211 
2.87633 
2.88056 
2.88479 
2.88904 
2.89330 
2.89756 
2.90184 
2.90613 


2.91042 
2.91473 
2.91904 
2.92337 
2.9277 

2.93204 
2.93640 
2.94076 
2.94514 


2.94952 |: 


2.95392 
2.95882 
2.96274 
2.96716 
2.97160 
2.97604 
2.98050 
2.98497 
2.98944 
2.99393 


2.99843 
3.00293 
3.00745 
3.01198 
8.01652 
3.02107 
3.02563 
3.03020 
3.08479 
3.03938 


3.04398 
3.04860 
3.05322 
3.05786 
3.06251 
8.06717 
3.07184 
3.07652 
3.08121 
8.08591 


3.09063 
3.09585 
8.10009 
8.10484 
8.10960 
3.11437 
3.11915 
3.12394 
3.12875 
3.138357 











he | 
woe SCOOVWSUR WOR OS 





= 
SOMOIHUR WMHS | 


76° 


Vers. | Ex. sec. 


.75808 | 8.13857 
75836 | 3.13889 
. 75864 | 3.14828 
75892 | 3.14809 
75921 | 3.15295 
75949 | 3.15782 
.T5977 | 3.16271 
.76005 | 3.16761 
76034 | 3.17252 
.76062 | 3.17744 
.76090 | 3.18238 
.76118 | 3.18733 
76147 | 3.19228 
V6175 | 8.19725 
76208 | 3.20224 
76231 | 3.20723 
76260 | 3.21224 
76288 | 8.21726 
£76316 | 3.22229 
. 76344 | 8.22734 
.76373 | 3.28239 


76401 | 3.23746 
76429 | 3.24255 
.76458 | 3.24764 
76486 | 8.25275 
76514 | 3.25787 
. 76542 | 3.26300 
.76571 | 8.26814 
.76599 | 8.27380 
76627 | 3.27847 
.76655 | 8.28366 


76684 | 3.28885 
76712 | 3.29406 
76740 | 3.29929 
76769 | 3.80452 
6797 | 8.80977 
76825 | 8.31503 
76854 | 3.32031 
.76882 | 3.32560 
.76910 | 3.388090 
. 76938 | 3.88622 


76967 | 3.34154 
76995 | 3.34689 
77023 | 3.85224 
77052 | 3.35761 
77080 | 8.36299 
77108 | 3.36839 
77137 | 3.87380 
77165 | 3.87923 
(7193 | 8.38466 
77222 | 3.39012 


77250 | 8.39558 
77278 | 3.40106 
773807 | 3.40656 
77335 | 3.41206 
77363 | 3.41759 
. (7392 | 3.42312 
77420 | 3.42867 
77448 | 8.43424 
JTIATT | 3.48982 
77505 | 3.44541 



























































G7 78° 

Vers. | Ex. sec.|| Vers. |Ex. sec. 
"7505 | 3.44541 79209 | 3.80973 
77589 | 3.45102 79237 | 3.81633 
77562 | 3.45664 .79266 | 3.82294 
77590 | 3.46228 . 79294 | 3.82956 
77618 | 3.46793 79828 | 3.88621 
77647 | 3.47360 .79351 | 8.84288 
77675 | 8.47928 || .793880 | 3.84956 
77703 | 3.48498 || .79408 | 3.85627 
77732 | 3.49069 .79437 | 8.86299 
.77760 | 8.49642 .79465 | 3.86973 
77788 | 3.50216 .79498 | 8.87649 
T7817 | 8.50791 || .79522 | 3.883827 
.77845 | 3.51368 . 79550 | 8.89007 
“7874 | 3.51947 21957 3.89689 
~77902 | 3.52527 .79607 | 3.903873 
77930 | 3.538109 .79636 | 8.91058 
77959 | 8.53692 .79664 | 8.91746 
W7987 | 3.54277 .79693 | 8.92486 
78015 | 8.54863 .79721 | 3.93128 
.78044 | 3.55451 79750 | 8.98821 
.78072 | 3.56041 ONT 3.94517 
.78101 | 3.56632 79807 | 3.95215 
678129 | 8.57224 .79835 | 3.95914 
.78157 | 3.57819 79864 | 3.96616 
.78186 | 3.58414 .79892 | 3.97320 
.78214 | 8.59012 .79921 | 3.98025 
. 78242 | 3.59611 79949 | 3.98783 
.78271 | 3.60211 .79978 | 3.994438 
.78299 | 3.60813 .80006 | 4.00155 
.78828 | 3.61417 .80035 | 4.00869 
.78356 | 8.62023 .80063 | 4.01585 
. 78384 | 3.62630 .80092 | 4.02803 
.78418 | 8.632388 .80120 | 4.03024 
.78441 | 3 63849 .80149 | 4.03746 
.78470 | 8.64461 80177 | 4.04471 
. 78498 | 8.65074 .80206 | 4.05197 
.78526 | 8.65690 ,80234 | 4.05926 
.78555 | 8.66307 || .80263 | 4.06657 
.78583 | 3.66925 .80291 | 4.07390 
.78612 | 8.67545 .80320 | 4.08125 
.78640 | 3.68167 || .80348 | 4.08863 
.78669 | 3.68791 || .80377 | 4.99602 
.78697 | 3.69417 .80405 | 4.10844 
. 78725 | 8.70044 .80434 | 4.11088 
78754 | 8.70673 ,80462 | 4.11885 
. 78782 | 3.718038 .80491 | 4.12588 
.78811 | 3.719385 .80520 | 4.13834 
.78839 | 3.72569 80548 | 4.14087 
78868 | 3.73205 8057 4.14842 
.78896 | 8.73843 80605 | 4.15599 
.78924 | 8.74482 |; .80684 | 4.16359 
.78958 | 3.75123 || .80662 | 4.17121 
78981 | 3.75766 .80691 | 4.17886 
.79010 | 8.76411 .80719 | 4.18652 
.79088 | 3.77057 .80748 | 4.19421 
.79067 | 3.77705 80776 | 4.20198 
.79095 | 3.78355 .80805 | 4.20966 
.79123 | 3.79007 .80883 | 4.21742 
.79152 | 3.79661 .80862 | 4.22521 
.79180 | 3.80316 .80891 | 4.23301 
.79209 | 3.8097 .80919 | 4.24084 





TABLE XXIX.—NATURAL VERSED SINES 


Vers. 


80919 
.80948 
.80976 
.81005 
.81083 
.81062 
.81090 
81119 
.81148 
81176 
81205 
.81233 
81262 
.81290 
81319 
.81348 
.81376 
.81405 





81433 
.81462 
81491 


.81519 
.81548 
81576 
81605 
.81633 
.81662 
.81691 
81719 
81748 
81776 
.81805 
.818384 
.81862 
.81891 
.81919 
.81948 
81977 
| .82005 
82084 
82068 


82091 
.82120 
82148 
82177 
82206 
82284 
82263 
82292 
82820 
82349 
82377 
82406 
82435 
82463 
82492 
82521 

82549 
82578 
82607 
82635 











79° 


Ex. sec. 


4.24084 
4.24870 
4.25658 
4.26448 
4.27241 
4.28036 
4.28633 
4.29634 
4.30486 
4.31241 
4.82049 


4.82859 
4.33671 
4.34486 
4.35304 
4.36124 
4.36947 
4.37772 
4.388600 
4.39430 
4.40263 


4.41099 
4.41987 
4.42778 
4.43622 


4.44468 - 


4.45317 
4.46169 
4.47028 
4.47881 
4.48740 


4.49603 
4.50468 
4.51337 
4.52208 
4.538081 


4.58958 | 


4.54837 
4.55720 
4.56605 
4.57493 


4.58383 
4.59277 
4.60174 


4.61073 | 


4.61976 
4.62881 
4.63790 
4.64701 
4.65616 
4.66533 


4.67454 
4.68877 
4.69304 
4.70284 
4.71166 
4.72102 
4.73041 
4.73983 
4.74929 
4.75877 





Somrounwnrnoe | 


rN 
ai 





. 


il 


AND EXTERNAL SECANTS 





™ 


Vers. 


82635 


CHMIBEA WMHS | 


10 














80° 





Ex. sec. 


| 4.75877 


4.76829 
4.77784 
4, 78742 
4.79703 


4.80667 


4.81635 
4.82606 
4.83581 
4.84558 
4.85539 


4.86524 
4.87511 
4.88502 
4.89497 
4.90495 
4.91496 
4.92501 
4.93509 
4.94521 
4.95536 
4.96555 
4.97577 


4.98603 | 


4.99633 
5.00666 
5.01703 
5.02743 
5.03787 
5.04834 
5.05886 


5.06941 
5.08000 
5.09062 
5.10129 
5.11199 
5.12278 
5.13350 
5.144382 
5.15517 
5.16607 
5.17700 
5.18797 
5.19898 
5, 21004 
5.22113 
5.23226 
5.24343 
5.25464 
5.26590 


“5.27719 


5.28853 
5.29991 
5.31133 
5.32279 
5.33429 
5.34584 
5.385743 
5.36906 
5.38073 
5.39245 














81° 


Vers. 


84357 
84385 
84414 
84443 
84471 
84500 
84529 
84558 
84586 
84615 
84644 


84673 
84701 
84730 
84759 
84788 
.84816 
84845 
84874 
84903 
84931 


.84960 
.84989 
85018 
85046 
85075 
85104 
85133 
85162 
85190 
85219 
85248 
85277 
85305 
85334 
85363 
85392 
85420 
85449 
85478. 
85507 


85536 
85564 
85593 
85622 
85651 
85680 
85708 
85737 
85766 
85795 


. 85823 


85852 


; .85881 


85910 
.85939 
85967 
85996 
86025 
86054 
86083 


Ex, sec. 


5.89245 
5.40422 
5.41602 
5.42787 
5.48977 
5.45171 
5.46369 
5.47572 
5.48779 
5.49991 
5.51208 


5.52429 
5.53655 
5.54886 
5.56121 
5.57361 
5.58606 
5.59855 
5.61110 
5.623869 
5.63633 


5.64902 
5 66176 
5.67454 
5.68738 
5.70027 
5.71821 
5.72620 
5.73924 
5.75233 
| 9.76547 


5.77866 
5.79191 
5.80521 
5.81856 
5.83196 
5.84542 
5.85893 
5.87250 
5.88612 
5.89979 


5.91352 
5.92731 
5.94115 
5.95505 
5.96900 
5.98301 
5.99708 
6.01120 
6.02538 
6.03962 
6.05392 








6.06828 


6.08269 
6.09717 
6.11171 
6.12630 
6.14096 
6.15568 
6.17046 
6.18530 














| 


Vers. 





86083 
.86112 
.86140 
.86169 
.86198 
86227 
86256 
86284 
86313 
86342 
86371 


86400 
86428 
86457 
86486 
86515 
86544 
86572 
.86601 
. 86630 
86659 


.86688 
86717 
86746 
86774 
. 86803 
86832 
86861 
. 86890 
86919 
86947 


86976 
87005 
87084 
.87063 
87092 
87120 
87149 
87178 
87207 
87285 
87265 
87294 
87822 
87351 
87380 
87409 
87438 
87467 
87496 
87524 
87553 
87582 
87611 
.87640 
87669 
87698 
87726 
87755 
87784 
87813 


82° 


Ex. sec. 


6.20020 
6.21517 
6.23019 
6.24529 
6.26044 
6.27566 
6.29095 
6.30680 
6.32171 
6.33719 


6.35274 
6.36835 
6.38403 
6.39978 
6.41560 
6.43148 
6.44743 
6.46346 
6.47955 
6.49571 


6.51194 
6.52825 
6.54462 
6.56107 
6.57759 
6.59418 
6.61085 
6.62759 
6.64441 
6.66130 


6.67826 
6.69530 
6.71242 
6.72962 
6.74689 
6. 76424 
6.78167 
6.79918 
6.81677 
6.83443 


6.85218 


7.20551 


6.18530 











83° 


Vers. 


.87818 
87842 
87871 
.87900 
.8792 

.87957 
87986 
.88015 
88044. 
.88073 
-88102 
.88131 


"89518 
‘89547 





Ex, sec. 


7.40466 


7.42511 
7.44566 
7.46632 
7.48707 


7.59241 
7.61379 
7 63528 
7.65688 
7.67859 
7.70041 
7.72234 
7.74438 
7.76653 
7.78880 
7.81118 
7.83367 


7 85628 
7.87901 
7.90186 
7.92482 
7.94791 
7.97111 


7.99444 |: 


8.01788 
8.04146 
8.06515 


8.08897 
8.11292 
8.138699 
8.16120 
8.18553 
8.20999 

238459 

20981 
8.28417 
8.30917 


| 8.334380 


8.85957 
8.38497 
8.41052 
8.43620 
8.46203 
8.48800 
8.51411 
8.54037 
8.56677 





_ 
SOOIND TP WOW O 


Sean e  rTEITnnEIEnEDEIE ESRI Ta 


299 TABLE XXIX.—NATURAL VERSED SINES 





Pt hed eet be 
He 09 0D Somaiammnono| 


15 














84° 

Vers, | Ex. sec, 
.89547 8.56677 
8957 8.59332 
. 89605 8.62002 
.89634 8.64687 
89663 8.67387 
89692 8.70103 
89721 8.72833 
89750 8.7557 

8977 8.78341 
89808 8.81119 
89836 8.83912 
89865 8.86722 
89894 8.89547 
89923 8.92889 
89952 8.95248 
89981 8.98123 
90010 9.01015 
90039 9.038923 
90068 9.06849 
90097 9.09792 
90126 9.12752 
90155 9.15730 
90184 9.18725 
90213 9.21739 
90242 9.24770 
90271 9.27819 
90300 9.30887 
90329 9.33973 
90358 9.37077 
90386 9.40201 
90415 9.438343 
90444 9.46505 
90473 9.49685 
90502 9.52886 
90531 9.56106 
90560 9.59346 
. 90589 9.62605 
.90618 9.65885 
. 90647 9.69186 
. 9067 9.72507 
.90705 9.75849 
.90734 9.79212 
.90763 9.82596 
. 90792 9.86001 
90821 9.89428 
. 90850 9.92877 
.90879 9.96348 
. 90908 9.99841 
. 90937 10.03356 
. 90966 10.06894 
. 90995 10.10455 
.91024 10.140389 
. 91053 10.17646 
.91082 10.21277 
91111 10. 24932 
91140 10. 28610 
.91169 10.32313 
.91197 10.86040 
. 91226 10.39792 
91255 10.43569 
. 91284 10.47371 














85° 

Vers. Ex. sec, 
91284 10.47371 
91813 10.51199 
91342 10.55052 
91371 10.58932 
91400 10.62837 
91429 10.66769 
91458 10. 70728 
91487 10.74714 
91516 10.78727 
91545 10.82768 
91574 1C 36837 
91603 10.90934 
916382 10.95060 
91661 10.99214 
91690 11.08397 
91719 11.07610 
.91748 11.11852 
91777 11.16125 
91806 11 .20427 
91835 11.24761 
91864 11.29125 
91893 11.33521 
91922 11 .37948 
91951 11.42408 
91980 11.46900 
92009 11.5142 

92038 11 .55982 
92067 11.6057% 
92096 11.65197 
92125 11.69856 
92154 11.74550 
92183 11.79278 
92212 11.84042 
92241 11.88841 
92270 11.938677 
92299 11.98549 
92328 12.03458 
92357 2.08404 
.92886 12.18388 
, 92415 12-18411 
92444 12.2847 
92473 12.28572 
. 92502 12.338712 
, 92581 12.38891 
. 92560 12.44112 
. 92589 12.49373 
. 92618 12.54676 
.92647 |. 12.60021 
. 9267 12.65408 
92705 12.70888 
927 12.76312 
92763 1281829 
92792 12.87391 
92821 12. 92999 
92850 12.98651 
9287 18 04350 
92908 13.10096 
92937 13.15889 
92966 13. 21730 
92995 13.27620 
93024 13.33559 














Vers. 


938024 


93053 
-93082 
93111 
.93140 
.93169 
.93198 
93227 
93257 
93286 
98815 


93344 
93373 
93402 
93431 
93460 
93489 
93518 
93547 
9357 

93605 


93634 
93663 
93692 
93721 
.93750 
9377 
98808 
93837 
. 93866 


*. 93895 


93924 
93953 
93982 
-94011 
.94040 
.94069 
. 94098 
94127 
.94156 
- 94186 


94215 
94244 
94273 
94302 
94331 
.94360 
.94389 
94418 
94447 
94476 


86° 


Ex. sec. 





13.83559 
13.389547 
13.45586 
13.51676 
13.57817 
13.64011 
13. 70258 
13.76558 


13.82913 


13.89323 
13.95788 
14.02310 
14.08890 
14. 15527 
14. 22223 
14.28979 
14.35795 
14. 42672 
14.49611 
14.56614 
14.63679 


14.70810 
14.78005 
14. 85268 
14.92597 
14,99995 
15.07462 
15, 14999 
15. 22607 
15.30287 
15.38041 


15, 45869 
15,58772 
15.61751 
1569808 
15. 77944 
15.86159 
15. 94456 
16 02835 
16.11297 
16, 19843 


16. 28476 
16.37196 
16.46005 
16.54903 
16 63893 
16.72975 
16 82152 
16,91424 
1700794 
17. 10262 


17.19830 
1729501 
17, 39274 
1749153 
17,59139 
1769233 
17.79438- 
1789755 
1800185 
18. 10732 








AND EXTERNAL SECANTS 





‘5 





, 


18. 10732 


18 .21397 
1832182 
18, 48088 
18.54119 
18.65275 
18.76560 
18 87976 
18.99524 
19.11208 
1923028 


19.34989 
19.47093 
19.59341 
19.71737 
19.84283 
19.96982 


. 20.09838 


20. 22852 
20 .36027 
20.49368 


20. 62876 
20.76555 
20.90409 
21.04440 
21.18653 
21.38050 
21.47635 
21.62413 
21.77386 
21.92559 
22.07935 
2223520 
22.39316 
2255329 
22.71563 
22. 88022 
23.04712 
23.216387 
23. 38802 
23,56212 


23, 73873 
23.91790 
24.09969 
24 28414 
24. 47134 
24.66182 
24. 85417 
25 04994 
25 24869 
25.45051 


25 65546 
25. 86360 
26 .07503 


26. 28981 
56. 50k04 
26 .72978 
26 95513 
27.18417 
27.41700 
27.65371 


97353 
97382 
97411 
.97440 
97470 
.97499 
97528 
97557 
97586 
97615 
97644 
97673 


88° 


Ex. sec, 


27 65371 
2789440 
28.18917 
28.38812 
28 64137 
28. 89903 
29.16120 
29 42802 
29.69960 
29.97607 
-80.25758 
30.54425 
80.83623 
31.13366 
31.48671 
31.74554 
32.06030 
32.38118 
382. 70835 
83.04199 
33. 388232 


33. 72952 
24,.08380 
34.44539 
34.81452 
35.19141 
3557633 
35. 96953 
36.37127 
36. 78185 
37. 20155 
37.63068 
38..06957 
3851855 
38.9797 
39. 44820 
39.92963 
40 42266 
40.9277 
41 44525 
41.97571 


42.51961 
43 07746 
43 .64980 
44 23720 
44 84026 
45 .45963 
46 .09596 
4674997 
47 42241 
48 .11406 


48 . 82576 
49 .55840 
50.31290 
51.09027 
51.89156 
52.71790 
53.57046 
54.45053 
55.35946 
56. 29869 























89° 
a 
Vers. Ex. sec, 
. 98255 56 .29869 0 
. 98284 57 .26976 1 
.98313 58 .27481 2 
. 98342 59.31411 3 
98371 60.39105 4 
.98400 61.50715 5 
. 98429 62 .66460 6 
.98458 63 .86572 7 
.98487 65.11304 8 
. 98517 66 .40927 9 
98546 67.75736 | 10 
.98575 69.16047 | 11 
. 98604 70.62207 | 12 
. 98633, %2.14583 13 
. 98662 3.78586 | 14 
- 98691 75 .89655 15 
.98720 "7 18274 16 
.98749 78 .94968 aly 
98778 80.85315 18 
. 98807 82.84947 19 
98836 84.94561 | 20 
.98866 87.14924 | 21 
. 98895 89. 46886 22 
. 98924 91. 91387 23 . 
.98953 94 .49471 24 
. 98982 97 .228038 25 
-99011 100:1119 26 
.99040 108.1757 | 27 
.99069 106.4811 28 
. 99098 109.8966 29 
. 99127 113.5930 30 
.99156 117.5444 | 31 
.99186 121.7780 32 
.99215 126.3253 33 
. 99244 131.2223 34 
. 99273 136.5111 85 
. 99302 142.2406 36 
. 99831 148.4684 37 
.99860 155.2623 38 
. 99889 162.7033 39 
.99418 170.8883 | 40 
.99447 179.9350 | 41 
.99476 189.9868 42 
. 99505 201 .2212 43 
. 99535 213.8600 44 
. 99564 228 .1889 45 
.99593 244 5540 46 
. 99622 263 4427 47 
.99651 285.4795 | 48 
. 99680 811.5280 | 49 
.99709 842.7752 | 50 
99738 880.9723 | 51 
.99767 428.7187 52 
. 99796 490.1070 53 
. 99825 571.9581 54 
.99855 686.5496 55 
. 99884. 858.4869 56 
.99913 1144.916 57 
. 99942 1717. 874 58 
. 99971 3436.747 59 
1.00000 Infinite 60 

















994 TABLE XXX.—CU.YDS. PER 100 FT. SLOPES 4:1 





Depth Base | Base | Base Base | Base | Base | Base | Base 

















d 12 14 16 18 22. | 24 26 28 
1 45 53 60 68 82 90 | 97 105 
2 93 107 122 137 167 181 196 211 
3 142 163 186 208 253 205 297 319 
4 193 222 252 281 341 37 400 430 
5 245 282 319 356 431 468 505 542 
6 300 344 389 433 522 567 611 656 
ff 356 408 460 512 616 668 719 T71 
8 415 474 533 593 711 wi 830 889 
9 475 542 608 675 808 875 942 1008 

10 537 611 685 759 907 981 1056 11380 

11 601 682 764 845 1008 1090 1171 1253 

12 667 756 844 933 ibhG! 1200 1289 1378 

13 734 831 926 1023 1216 1312 1408 1505 

14 804 907 1010 1115 1322 1426 1530 1633 

15 875 986 1096 1208 1431 1542 1653 1764 

16 948 1067 1184 1804 1541 1659 1778 1896 

17 1023 1149 127 1401 1653 177 1905 2031 

18 1100 1233 1366 1500 1767 1900 2033 2167 

19 1179 1319 1460 1601 1882 2028 2164 2305 

20 1259 1407 1555 1704 2000 2148 2296 2444 

21 1342 1497 1653 1808 2119 2275 2431 2586 

22 1426 1589 1752 1915 2241 2404 2567 2730 

23 1512 1682 1853 2023 2364 2534 705 287; 

24 1600 177 1955 2138 2489 2667 2844 8022 

25 1690 1875 2060 2245 2616 2801 2986 3171 

26 1781 197 2166 2359 2744 2987 8130 8322 

27 1875 2075 2274 2475 2875 8075 827 B47 

28 1970 2178 2384 2593 3007 3215 8422 3630 

29 2068 2282 2496 2712 3142 3358 857 3786 

80 2167 2889 2610 2833 8278 3500 3722 3944 

31 2268 2497 2726 2956 3416 3645 387 4105 

32 2370 2607 2844 3081 3556 3793 4020 4267 

383 2475 2719 2964 8208 3697 3942 4186 4431 

34 2581 2833 8085 3337 3841 4093 4344 4596 

35 2690 2949 3208 3468 3986 4245 4595 4764 

36 2800 8067 3338 3600 4138 4400 4667 4933 

37 2912 3186 3460 3734 4282 4556 4831 5105 

88 3026 8807 3589 387 4433 4715 4996 527 

39 3142 3431 3719 4008 4586 487 5164 5453 

40 8259 8556 8852 4148 4741 5037 5333 5630 

41 3379 3682 3986 4290 4897 5201 5505 5808 

42 8500 3811 4122 4433 5056 5367 5678 5989 

43 8623 8942 4260 4579 5216 5534 5853 6171 

44 3748 4074 4400 4726 5378 5704 6030 6356 

45 387 08 4541 4875 5542 5875 6208 6542 

46 4004 4344 4684 5026 5707 6048 6389 6730 

47 4134 4482 4830 5179 5875 6223 6571 6919 

48 4267 4622 4978 5333 6044 6400 6756 7111 

49 4401 4764 5127 5490 6216 6579 6942 7305 

50 4537 4907 5278 5648 6389 6759 7130 7500 

51 4675 5053 54380 5808 6564 6942 7319 7697 

52 4815 5200 5084 5970 6741 7126 7511 7896 

53 4956 5349 5741 6134 6919 7312 7705 8097 

54 5100 5500 5900 6300 7100 7500 7900 8300 

55 5245 5653 6060 6468 7282 7690 8097 8505 

56 5393 5807 6222 6637 7467 7881 8296 8711 

57 5542 5964 6386 6808 7653 8075 8497 8919 

58 5693 6122 6552 6981 7841 8270 8700 9130 

59 5845 6282 6719 7156 8031 8468 8905 9342 








Ed 


TABLE XXX.—CU.YDS. PER 100 FT. SLOPES 4:1 225 


Depth | Base | Base | Base | Base | Base | Base | Base | Base 











d 12 i 16 18 22 24 26 28 
1 46 bd 61 69 83 91 98 |- 106 
2 G6 fopeyit +} 186°)| West [ets 185 | 200 | 215 
3 150 72 | 194 | 217 |. 261 | 283 | 306 | $28 
4 207 | 237 | 267 | 296 | 356 | 88 | 415 | 444 
5 969 | 306 | 343 | 880 | 454 | 491 | 528 | 565 
6 333 | 8y 422 | 467 | 556 | 600 | 644 | 689 
7 402 | 454 | 506 | 557 | 661 | 713 | %65 | 817 
8 474-| 533 | 593 | 652 | 70 | 830 | 899 | 948 
9 550 | 6l7 | 683 | 750 | 883 | 950 | 1017 | 1088 

10 630 | 704 | 778 | 852 { 1000 | 1074 | 1148 | 1222 

11 713 | 94 | 976 | 957 | 1120 | 1202 | 1283 | 1365 

12 s00 | 889 } 978 | 1067 | 1244 | 1333 | 1422 | 1511 

13 91 | 987 | 1083 | 1180 |, 1372 | 1469 | 1565 | 166% 

14 985 | 1089 | 1193 | 1296 |' 1504 | 1607 | 1711 | 1815 

15 1083 | 1194 | 1306 | 1417 | 1639 | 1750 | 1861 | 1972 

16 1185 | 1304 | 1422 | 1541 | 1779 | 1896 | 2015 | 2138 

17 1291 | 1417 | 1543 | 1669 | 1920 | 2046 | 2172 | 2298 

18 1400 | 1533 | 1667 | 1800 | 2067 | 2200 | 2333 | 2467 

19 | ~1513 | 1654 | 1794.) 1985 | 2217 | 2857 | 2498 | 2639 

20 1630.| 1778 | 1926 | 2074 | 2370 | 2519 | 2667 | 2815 

21 1750 | 1906 | 2061 | 2217 | 2528 | 2683 | 2889 | 2904 

22 1874 | 2037 | 2200 | 2363 | 2689 | 2852 | 3015 | 3178 

23 2002 | 2172 | 2343 | 9513 | 2854 | 3024 | 3194 | 3365 

24 2133 | 2311 | 2489 | 2667 | 3022 | 3200 | 3378 | 3556 

25 9269 | 2454 | 2639 | 2824 | 3194 | 3380 | 3565 | 3750 

26 2407 | 2600 | 2793 | 2985 | 3370 | 3563 | 3756 | 3948 

27 9550 | 2750 | 2950 | 3150 | 3550 | 3750 | 3950 | 4151 

28 2696 | 2904 | 3111 | 3319 | 3733 | 3941 | 4148 | 4356 

29 2846 | 3061 | 3276 | 3491 | 3920 | 4185 | 4350 | 4565 

30 3000 | S222 | 3444 | 3667 | 4111 | 4333 | 4556 | 4778 

31 3157 | 3387 | 3617 | 3846 | 4306 | 4535 | 4765 | 4994 

32 3319 | 3556 | 3793 | 4030 | 4504 | 4741 | 4978 | 5215 

33 3483 | 3728 | 3072 | 4217 | 4706 | 4950 | 5194 | 5439 

“34 3652 | 3004 | 4156 | 4407 | 4911 | 5163 | 5415 | 5667 

35 3824 | 4083 | 4343 | 4602 | 5120 | 5380 | 5639 | 5898 

36 4000 | 4267 | 4533 | 4800 | 5333 | 5600 | 5867 | 6133 

37 4180 | 4454 | 4728 | 5002 | 5550 | 5824 | 6098 | 6372 

38 4363 | 4644 | 4926 | 5207 | 5770 | 6052 | 6333 | 6615 

39 4550 | 4839 | 5128 | 5417 | 5994 | 6283 | 6572 | 6861 

40 4741 | 5u87 | 5833 | 5630 | G2a2 | 6519 | 6815 | 7111 

41 4935 | 5239 | 5543 | 5846 | 6454 | 6757 | OGL | 7365 

42 5133 | 5444 | 5756 | 6067 | 6689 | 7000 | YBil | 7622 

48 5385 | 5654 | 5972 | 6201 | 6928 | 7246 | 7565 | 7883 

44 5541 | 5867 | 6193 | 6519 | 7170 | 7496 | 7822 | 8148 

45 Bv50 | 6083 | 6417 | 6750 | 7417 | 7750 | 8083 | 8417 

46 5963 | 6304 | 6644 | 6985 | Ye67 | S007 | 8348 | 8689 

47 6180 | 6528 | 6876 | 7224 | 7920 | 8269 | 8617 | 8965 

48 6400 | 6756 | T11l | 7467 | 8178 | 8533 | 8889 | 9244 

49 6624 | 6987 | 7350 | 7713 | 8439 | 8802 | 9165 | 9528 

50 6852 | 7222 | 7593 | 7963 | 8704 } 9074 | 9444 | 9815 

Bt oss | 7461 | 7839 | S217 | so72 | 9350 | 9728 | 10106 

52 7319 | 7704 | 8089 | 8474 | 9244 | 9630 | 10015 | 10400 

53 7557 | 7950 | 8343 | 8735 | 9520 | 9913 | 10306 | 10698 

54 7800 | 8200 | 8600 | 9000 | 9800 | 10200 | 10600 | 11000 

55 8046 | 8454 | 8561 | 9269 | 10083 | 10491 | 10898 | 11306 

56 9296 | 8711 | 9126 | 9541 | 10370 | 10785 | 11200 | 11615 

Bz 8550 | 8972 | 9394 | 9817 | 10661 | 11083 | 11506 | 11928 

5 s807 | 9237 | 9667 | 10096 | 10956 | 11385 | 11815 | 12244 

59 9069 | 9506 | 9943 | 10380 | 11254 | 11691 | 12198 | 12565 


226 


° 


TABLE XXX.—CU.YDS. PER 100 FT. 


SLOPES 1:1 





Depth 
d 


Base 
12 








Base 
14 


Base 
16 








Base 


18 








Base 


20 








Base 


28 


Base 


30 





Base 
32 





TABLE XXX.—CU.YDS. PER 100 FT. 


16 


10212 
10656 
11108 
11570 
12042 
12522 
13012 
13511 
14019 
14537 


15064 
15600 
16145 
16700 
17264 
17837 
18419 
19011 
19612 
20222 

















Base 
18 





10516 
10967 
11427 
11896 
12375 
12863 
13360 
13867 
14382 
14907 
15442 
15985 
16538 
17100 
17671 
18252 
18442 
19441 
20049 
20667 


Base 
20 











Base 
28 


12034 
12522 
13020 
13526 
14042 
14567 
15101. 
15644 
16197 
16759 


17331 
17911 
18501 
19100 
19708 
20326 
20953 
21589 
222384 
22889 





Base 
30 











SLOPES 1144:1 227 





18086 
18681 
19286 
19900 
20523 
21156 


*| 21797 


22448 
23108 





23778 
/ 





228 TABLE XXX.—CU.YDS. PER 100 FT. SLOPES 1144:1 





Depth Base | Base | Base | Base | Base | Base | Base | Base 














d 12 14 16 18 20 28 30 32 
1 50 57 65 (2 80 109 117 124 
2 i111 126 141 156 70 23 244 259 
3 183 206 228 250 272 361 3883 406 
4 267 296 |° 826 856 885 504 533 563 
5 361 898 435 472 509 657 694 731 
6 467 511 556 600 644 822 867 911 « 
7 583 635 687 739 791 998 1050 1102 
8 711 770 830 889 948 1185 1244 1304 
9 850 917 983 1050 1116 1383 1450 1517 

10 1000 1074 1148 1222 1296 1593 1667 1741 

11 1161 1248 1324 1406 1487 18138 1894 1976 

12 1333 1422 1511 1600 1689 2044 2133 2222 

13 1517 1613 709 1806 1902 2287 2383 2480 

14 ial 1815 1919 2022 212 2541 2644 2748 

15 1917 2028 218 2250 2361 2806 2917 8028 
16 2133 2252 237 2489 2607 3081 3200 8319 

17 2361 2487 2613 2739 2865 3369 3494 3620 

18 2600 2783 2867 38000 31383 3667 3800 3933 
19 2850 2991 3131 8272 3413 397 4117 4257 

20 3111 8259 3407 8556 3704 4296 ddd 4592 

21 3383 8539 3694 8850 4005 4628 4783 4939 

22 3667 8830 38993 4156 4318 4970 5183 5296 

28 3961 4131 4802 447 4642 5824 5494 5665 

24 4267 4444 4622 4800 4978 5689 5867 6044 

25 4583 4769 4954 5139 5324 6065 6250 6485 

26 4911 5104 5296 5489 5681 6452 6644 6837 

27 5250 5450 5650 5850 6050 6850 7050 7250 

28 5600 5807 6015 6222 6430 7259 7467 7674 

29 5961 6176 6391 6606 6820 7680 7894 8109 

30 6333 6556 6778 7000 7222 8111 8333 8555 

31 6717 6946 7176 7406 7635 8554 8783 9013 

32 7111 7348 7585 7822 8059 9007 9244 9482 

33 17 7761 8006 8250 8494 9472 T17 9962 

34 7933 8185 8437 8689 8941 9948 | 10200 | 10452 

35 8361 8620 8880 9139 9398 | 10435 | 10694 | 10954 

36 8800 9067 9333 9600 9867 | 109383 | 11200 | 11467 

37 9250 9524 9798 | 10072 | 10346 | 11443 | 11717 | 11991 

38 9711 9993 | 10274 | 10556 | 10837 | 11963 | 12244 | 12526 











TABLE XXX.—CU.YDS. PER 100 FT. 


Depth 


d 


£O OD =F Od OF He OD DW 





Base 


12 


4 
28407 
29333 


SLOPES 2:1 
































Base Base Base Base Base Base 
14 16 18 20 28 30 
59 67 ff 81 111 119 
133 148 163 178 237 252 
222 244 267 289 378 400 
326 356 385 415 533 563 
444 481 519 556 704 41 
578 622 667 711 889 933 
(26 U7 830 881 1089 1141 
889 948 1007 1067 1304 1363 
1067 1133 1200 1267 1583 1600 
1259 1333 1407 1481 1778 1852 
1467 1548 1630 1711 2037 2119 
1689 V7 1867 1956 2311 2400 
1926 2022 2119 2215 2600 2696 
217 2281 2385 2489 2904 3007 
2444 2556 2667 27°78 8222 3333 
2726 2844 2963 3081 3556 3674 
3022 3148 38274 3400 3904 4030 
5303 3467 3600 3733 4267 4400 
3659 3800 3941 4081 4644 A485 © 
4000 4148 4296 4444 5037 5185 
4356 4511 4667 4822 5444 5600 
4730 4889 5052 5215 5867 6030 
5111 5281 5452 5622 6304 6474 
5511 5689 5867 6044. 6756 6933 
5026 6111 6296 6481 7222 7407 
6356 6548 6741 6933 704. 7896 
6800 7000 7200 7400 8200 8400 
7259 7467 7674 7881 8711 8919 
7738 7948 8163 8378 9237 9452 
8222 8444 8667 8889 977 10000 
8726 8956 9185 9415 | 10333 10563 
9244 9481 9719 9956 10904 11141 
9778 | 10022 | 10267 | 10511 11489 lilies) 
10326 | 10578 | 10880 11081 12089 | 12841 
10889 | 11148 | 11407 11667 | 12704 12963 
11467 | 1173838 | 12000 12267 | 13838 13600 
12059 12833 | 12607 12881 13978 14252 
12667 12948 | 18230 13511 14637 14919 
18289 13578 | 18867 | 14156 15311 15600 
13926 14222 | 14519 14815 16000 | 16296 
14578 | 14881 15185 | 1548 16704 | 17007 
15244 | 15556 | 15867 | 16178 | 17422 | 17733 
15926 | 16224 | 16563 | 16881 18156 | 18474 
16622 | 16948 | 17274 | 17600 | 18904 | 19230 
17383 | 1766% | 18000 | 18333 19667 | 20000 
18059 | 18400 | 18741 19081 20444 | 20785 
18800 | 19148 | 19496 | 19844 | 21287 | 21585 
19556 | 19911 | 20267 | 20622 | 22044 | 22400 
20826 | 20689 | 21052 | 21415 | 22867 | 23230 
20711 21481 | 21852 | 22222 | 23704 | 2407 
21911 | 22289 | 22667 | 238044 | 24556 | 24933 
22726 | 23111 | 23496 | 23881 | 25422 | 25807 
23556 | 23948 | 24341 | 24733 | 26304 | 26696 
24400 | 24800 | 25200 | 25600 | 27200 | 27600 
25259 | 25667 | 2607 26481 28111 | 28519 
261383 | 26548 | 26963 7378 | 29087 -| 29452 
27022 | 27444: | 27867 | 28289 | 29978 | 30400 
27926. | 28356 | 28785 | 29215 | 309383 | 381363 
1 28844 | 29281 | 29719 | 80156 | 31904 | 32841 
| 29778 80222 | 30667 | 31111 | 82889 | 38333 








ee 


230 TABLE XXX.—CU.YDS. PER 100 FT. SLOPES3:1 





Depth | Base | Base | Base | Base | Base | Base | Base | Base 














d 12 14 16 18 20 28 30 32 
1 56 63 “ 78 85 115 122 130 
2 133 148 163 178 193 252 267 281 
3 233 256 27 300 822 411 433 456 
4 356 385 415 44 474 593 622 652 
5 500 537 74 611 648 796 833 870 
6 667 711 756 800 844 1022 1067 1111 
i$ 856 907 959 1011 1063 127 1822 1374 
8 1067 1126 1185 1244 1304 1541 1600 1659 
uy 1300 1367 1433 1500 1567 1833 1900 1967 

10 1556 1630 1704 1778 1852 2148 2222 2296 

11 1833 1915 1996 2078 2159 2485 2567 2648 

12 2183 2222 2311 2400 2489 2844 2933 8022 

13 2456 2552 2648 2744 2841 8226 3322 3419 

14 2800 2904 8007 8111 8215 3630 3733 3837 

15 3167 8278 3389 3500 3611 4056 4167 427 

16 8556 8674 3793 38911 4030 4504 4622 4741 
7 3967 4093 4219 4344 447 4974 5100 5226 

18 4400 4533 4667 4800 4933 5467 5600 5733 

19 4856 4996 5187 5278 5419 5981 6122 6263 

20 53833 5481 5630 5778 5926 6519 6667 6815 

21 5833 5989 6144 6300 6456 7078 7233 7389 

22 6356 6519 6681 6844 7007 7659 7822 7985 

28 6900 70% (241 7411 7581 8263 8433 8504 

24 7467 (644 7822 8000 8178 8889 9067 9144 

25 8056 8241 8426 8611 8796 9537 9722 9807 

26 8667 8859 9052 92 9437 | 10207 | 10400 | 10593 

27 9300 9500 9700 9900 | 10100 | 10900 | 11100 | 11300 

28 9956 | 10163 | 10370 | 10578 | 10785 | 11615 | 11822 | 12080 

29 10633 | 10848 | 11063 | 11278 | 11493 | 12852 | 12567 | 12781 

30 11883 | 11556 | 11778 | 12000 | 12222 | 18111 | 18383 | 18556 

31 12056 | 12285 | 12515 | 12744 | 12974 | 138893 | 14122 | 14852 





382 12800 | 13087 | 18274 | 18511 | 18748 | 14696 | 149383 | 15170 
- 383 13567 | 13811 | 14056 | 14300 | 14544 | 15522 | 15767 | 16011 
34 14356 | 14607 | 14859 | 15111 | 15363 | 1637 16622 | 16874 
35 15167 | 15426 | 15685 | 15944 | 16204 | 17241 | 17500 | 17759 
36 16000 | 16267 | 16533 | 16800 | 17067 | 18183 | 18400 | 18667 
37 16856 | 17180 7404 | 17678 | 17952 | 19048 | 19322 | 19596 
38 17783 | 18015 | 18296 | 1857 18859 | 19985 | 20267 | 20548 
39 18683 | 18922 | 19211 | 19500 | 19789 | 20944 | 21233 | 21522 
40 19556 | 19852 | 20148 | 20444 | 20741 | 21926 | 22222 | 22516 


4} 20500 | 20804 | 21107 | 21411 | 21715 | 229380 | 28283 | 23537 
42 21467 | 21778 | 22089 | 22400 | 22711 | 28956 | 24267 | 24578 
43 22456 | 22774 | 23093 | 28411 | 237380 | 25004 | 25322 | 25641 
44 23467 | 238793 | 24119 | 24444 | 24770 | 26074 | 26400 | 26726 
45 24500 | 24883 | 25167 | 25500 | 25833 | 27167 | 27500 | 27833 
46 25556 | 25896 | 26237 | 26578 | 26919 | 28281 | 28622 | 28963 
47 26633 | 26981 | 27330 | 27678 | 28026 
48 ; 28089 | 28444 | 28800 | 29156 | 30578 | 30933 | 31289 
49 28856 | 29219 | 29581 | 29944 | 303807 t 

50 80000 | 803870 | 380741 | 81111 | 381481 | 32963 | 33333 | 38704 


51 81167 | 31544 | 31922 | 82300 | 32678 | 34189 | 34567 | 34944 
52 823856 | 82741 | 83126 | 33511 | 33896 | 85487 | 35822 | 36207 
53 33567 | 838959 | 34352 | 34744 | 85137 | 86707 | 387100 | 87493 
54 84800 | 35200 | 35600 | 36000 | 36400 | 388000 | 388400 | 38800 
55 36056 | 386463 | 36870 | 387278 | 37685 | 39315 | 39722 | 40130 
56 37333 | 87748 | 38163 | 88578 | 88993 | 40652 | 41067 | 41481 
57 38633. | 39056 | 39478 | 39900 | 40322 | 42011 | 42433 | 42856 
58 89956 | 40385 | 40815 | 41244 | 41674 | 43393 | 48822 | 44252 
59 41300 | 417387 | 42174 | 42611 | 438048 | 44796 | 45233 | 45670 
60 42667 | 48111 | 48556 | 44000 | 44444 | 46222 | 46667 | 47111 





S 
3 
ies) 
co 
BSS 
30TH 
Orstee 
OW O 
09 G9 
wow 
rt ( 2 
wg a 
WOON 
Oo 

ae 
Ore Or 











XXXI.—TRIANGULAR PRISMS. CU.YDS. PER 50 FT.. 231 





















































WIDTH. 
» 
oO 
ee 1 2 3 4 5 6 7 8 
0.1 098 185 278 3870 463 566 648 . 741 
7g Nas Rts} 370 556 TAL 926 | 1.111 | 1.296} 1.481 
3 | .278 556 833 | 1.111 | 1.389 | 1.667 | 1.944 | 2.222 
4 | .370 741 | 1.111 | 1.481 | 1.852 | 2.222 | 2.593 | 2.963 
5 463 926} 1.389 | 1.852 | 2.3815 | 2.778 | 3:241 | 3.704 
G-} 3506) Felt 1.667 | 2.222 | 2.778 | 3.38383 | 3.889 | 4.444 
7 | .648; 1.296 | 1.944 | 2.593 | 3.241 | 3.889 | 4.587 | 5.185 
8 | .741) 1.481 | 2.222 | 2.963 | 3.704 | 4.444 | 5.185 | 5.926 
9 | .833) 1.667 | 2.500] 3.333 | 4.167 | 5.000 | 4.8383 | 6.667 
1.0| .926) 1.852] 2.778 | 3.704) 4.630) 5.556} 6.481 | 17.407 
-1 | 1.019} 2.087 | 38.056 | 4.074 | 5.093 | 6.111 | 17.1380] 8.148 
v2 | LelId @ 2.222 |) 86838) 4.444 | 5.556 | 6.667") T-778)). 8.889 
.8 | 1.204) 2.407 | 38.611 | 4.815 | 6.019 | 7.222 | 8.426 | 9.630 
4 | 1.296 2.598 | 3.889 ; 5.185 | 6.481 | 7.778 | 9.074 ; 10.370 
.5 | 1.389} 2.778 | 4.167 | 5.556 | 6.944 | 8.333 | 9.722 | 11.111 
.6 | 1.481) 2.963 | 4.444 | 5.926 | 7.407} 8.889 | 10.370 | 11.852 
.7| 1.574 8.148 | 4.722 | 6.296 | 7.870 | 9.444 | 11.019 | 12.598 
78 | 1.667) 3.333 | 5.000 | 6.667 | 8.333 | 10.000 | 11.667 | 13.333 
.9 | 1.759) 8.519 | 5.278 | 7.037 | 8.796 | 10.556 | 12.315 | 14.074 
2-0 | 1.852). 3.704 | 5.556.) 7.407 | 9.259) 11-111} 12.963 | 14.815 
.1 | 1.944; 3.889 | 5.883 | 7.778 | 9.722 | 11.667 | 13.611 | 15.556 
.2 | 2.087) 4.074} 6.111 | 8.148 | 10.185 | 12.222 | 14.259 | 16.296 
.38 | 2.180! 4.259 | 6.389 | 8.519 | 10.648 | 12.778 | 14.907 | 17.037 
4 | 2.222) 4.444 | 6.667 | 8.889 | 11.111 | 18.333 | 15.556 | 17.778 
.5 | 2.815} 4.630} 6.944 | 9.259 | 11.574 | 13.889 | 16.204 | 18.519 
.6 | 2.407) 4.815 |. 7.222 | 9.630 | 12.087 | 14.444 |. 16.852 | 19.259 
.7 | 2.500] 5.000 | 7.500 | 10.000 | 12.500 | 15.000 | 17.500 | 20.000 
.8 | 2.593} 5.185 | 7.778 | 10.370 | 12.968 | 15.556 | 18.148 | 20.741 
.9 | 2.685} 5.370 | 8.056 | 10.741 | 18.426 | 16.111 | 18.796 | 21.481 
3.0 |. 2.778) 5.556 | 8.338 | 11.111 | 18.889 | 16.667 | 19.444 | 22.222 
.1 | 2.870) 5.741 | 8.611 | 11.481 | 14.852 | 17.222 | 20.093 | 22.963 
.2 | 2.968] 5.926 | 8.889 | 11.852 | 14.815 | 17.778 | 20.741 | 23.704 
.8 | 3.056} 6.111 | 9.167 | 12.222 | 15.278 | 18.333 | 21.389 | 24.444 
<4 | 3.148] 6.296 | 9.444 | 12.593 | 15.741 | 18.889 | 22.037 | 25.185 
.5 | 8.241) 6.481 | 9.722 | 12.963 | 16.204 | 19.444 | 22.685 | 25.926 
.6 | 3.333) 6.667 | 10.000 | 13.333 | 16.667 | 20.000 | 23.333 | 26.667 
.7 | 3.426) 6.852 | 10.278 | 18.704 | 17.180 | 20.556 | 23.981 | 27.407 
-.8 | 3.519) 7.087 |.10.556 | 14.074 | 17.593 | 21.111 | 24.6380 | 28.148 
.9 | 3.611) 7.222 | 10.833 | 14.444 | 18.056 | 21.667 | 25.278 | 28.889 
4.0 | 3.704, 7.407 | 11.111 | 14.815 | 18.519 | 22.222 | 25.926 | 29.630 
* .1 | 3.796) 7.598 | 11.389 | 15.185 | 18.981 | 22.778 | 26.574 | 30.370 : 
.2 | 3.889} 7.778 | 11.667 | 15.556 | 19.444 | 23.3383 | 27.222 | 31.111 : 
.8 | 3.981} 7.968 | 11.944 | 15.926 | 19.907 | 23.889 | 27.870 | 31.852 : 
-4| 4.074) 8.148 | 12.222 | 16.296 | 20.870 | 24.444 | 28.519 | 32.598 : 
.5 | 4.167; 8.338 ; 12.500 | 16.667 | 20.833 | 25.000 | 29.167 | 33.333 ; 
.6 | 42259} 8.519 | 12.778 | 17.087 | 21.296 | 25.556 | 29.815 | 34.074 
.7 | 4.3852) 8.704 | 18.056 | 17.407 | 21.759 | 26.111 | 30.468 | 34.815 : 
.8 | 4.444) 8.889 | 13.3833 | 17.778 | 22.222 | 26.667 | 81.111 | 35.556 : 
.9 | 4.5387] 9.074 | 18.611 | 18.148 | 22.685 | 27.222 | 31.759 | 36.296 
5.0 | 4.680} 9.259 | 13.889 | 18.519 | 23.148 | 27.778 | 32.407 | 37.037 : 
.1 | 4.722) 9.444 | 14.167 | 18.889 | 23.611 | 28.333 | 33.056 | 37.778 ; 
.2 | 4.815} 9.630 | 14.444 | 19.259 | 24.074 | 28.889 | 83.704 | 88.519 
.8 | 4.907| 9.815 | 14.722 | 19.630 | 24.587 | 29.444 | 34.352 | 39.259 .167> 
.4 | 5.000} 10.000 | 15.000 | 20.000 | 25.000 | 80.000 | 35.000 | 40.000 : 
.5 | 5.093} 10.185 | 15.278 | 20.370 | 25.463 | 80.556 | 35.648 | 40.741 ' 
.6 | 5.185} 10.370 | 15.556 | 20.741 | 25.926 | 81.111 | 36.296 | 41.481 : 
7 | 5.278) 10.556 | 15.833 | 21.111 | 26.389 | 31.667 | 36.944 | 42.222 ! 
.8 | 5.370) 10.741 | 16.111 | 21.481 | 26.852 | 32.222 | 37.593 | 42.963 : 
9 | 5.463) 10.926 | 16.389 | 21.852 | 27.315 | 32.778 | 38.241 | 43.704 167 
6.0 | 5.556) 11.111 16.667 | 22.222 | 27.778 | 33.333 | 38.889 | 44.444 
a 





TABLE XXXI.—TRIANGULAR PRISMS 


WIDTH. 





2 











11.296 


12.222 
12.407 


12.778 
12.963 


13.148 


14. 259 
14.444 


14.815 


15.000 
15.185 


15.556 
15.741 


16. 296 


16.667 
16. 852 


18.148 
18.333 
18.519 


18.704 


19.630 


19.815 


20.370 
20.556 


21.944 
e2.ec0 


22.500 


25.000 
20.278 


26.944 
21.202 


27.778 
28.056 








22.593 


23.333 
23.704 


24.444 
24.815 


25.556 
25.926 


26.296 


27.037 
27.407 


28.148 
28.519 


29. 630 


30.000 
30.370 


31.111 
31.481] 


33.333 
33.704 


36. 296 
36. 667 
37.037 


37.407 





5 





28.241 
28.704 
29.167 
29.630 
30.093 
30.556 
31.019 
31.481 
31.944 
32.407 


32.870 
33.333 
33.796 
34.259 
34.722 
35.185 
35. 648 
36.111 
36.574 
37.037 


37.500 
38. 426 


41.667 
42.130 


44.907 
45.370 


46. 296 
46.759 


50.000 
50. 463 
50.926 


51.389 























6 7 8 9 
33.889 | 39.5374} 45.185 | 50.833 
34.444 | 40.185 | 45.926 | 51.667 
35.000 | 40.833 | 46.667 | 52.500 
35.556 | 41.481 | 47.407 | 53.333 
36.111 | 42.130 | 48.148 | 54.167 
36.667 | 42.778 | 48.889 | 55.000 
37.222 | 43.426 | 49.630 | 55.833 
37.778 | 44.074 | 50.370 | 56.667 
38.333 | 44.722 | 51.111 | 57.500 
38.889 | 45.370 | 51.852 | 58.333 
39.444 | 46.019 | 52.593 | 59.167 
40.000 | 46.667 | 53.333 | 60.000 
40.556 | 47.315 | 54.074 | 60.833 
41.111 | 47.963 | 54.815 | 61.667 
41.667 | 48.611 | 55.556 | 62.500 
42.222 | 49.259 | 56.296 | 63.333 
42.778 | 49.907 | 57.037 | 64.167 
43.333 | 50.556 | 57.778 | 65.000 
43.889 | 51.204 | 58.519 | 65.833 
44.444 | 51.852 | 59.259 | 66.667 
45.000 | 52.500 | 60.000 | 67.500 
45.556 | 53.148 | 60.741 | 68.333 
46.111 | 53.796 | 61.481 | 69.167 
46.667 | 54.444 | 62.222 | 170.000 
47.222 | 55.093 | 62.963 | 70.833 
47.778 | 55.741 | 63.704 | 71.667 
48.333 | 56.389 | 64.444 | 72.500 
48.889 | 57.037 | 65.185 | 73.333 
49.444 | 57.685 | 65.926 | 174.167 
50.000 | 58.333 | 66.667 | 75.000 
50.556 | 58.981 | 67.407 | 75.833 
51.111 | 59.630 | 68.148 | 76.667 
51.667 | 60.278 | 68.889 | 77.500 
52.222 | 60.926 | 69.630 | 78.333 
52.778 | 61.574 | 70.370 | 79.167 
53.333 | 62.222 | 71.111 | 80.000 
53.889 | 62.870 | 71.852 | 80.833 
54.444 | 63.519 | 72.593 | 81.667 
55.000 | 64.167 | 73.333 | 82.500 
55.556 | 64.815 | 74.074 | 83.333 
56.111 | 65.463 | 74.815 | 84.167 
56.667 | 66.111 | 75.556 | 85.000 
57.222 | 66.759 | 76.296 | 85.833 
57.778 | 67.407 | 77.037 | 86.667 
58.333 | 68.056 | 77.778 | 87.500 
58.889 | 68.704 | 78.519 | 88.333 
59.444 | 69.352 | 79.259 | 89.167 
60.000 | 70.000 | 80.600 | 90.000 
60.556 | 70.648 | 80.741 | 90.833 
61.111 | 71.296 | 81.481 | 91.667 
61.667 | 71.944 | 82.222 | 92.500 
62.222 | 72.593 | 82.963 | 93.333 
62.778 | 73.241 | 83.704 | 94.167 
63.333 | 73.889 | 84.444 | 95.000 
63.889 | 74.537 | 85.185 | 95.833 
64.444 | 75.185 | 85.926 | 96.667 
65.000 | 75.833 | 86.667 | 97.500 
65.556 | 76.481 | 87.407 | 98.333 
66.111 | 77.130 | 88.148 | 99.167 
66.667 | 77.778 | 88.889 | 100.000 


\ 


CU: YDS. PER 50 FT. 





233 






































WIDTH. 
C | 
jen} 1 2 3 4. 5 6 ve 8 9 
12.1/11.204] 22.407 | 33.611 | 44.815 | 56.019 | 67.222 78.426] 89.630) 100.833 
-2/11.296| 22.593 | 33.889 | 45.185 | 56.481 | 67.778 79.074; 90.370) 101.667 
-3/11.389| 22.778 | 34.167 | 45.556 | 56.944 | 68.333 79.722) 91.111) 102.500 
-4\11.481] 22.963 | 34.444 | 45.926 | 57.407 | 68.889 80.370) 91.852) 108.333 
5/11.574| 23.148 | 34.722 | 46.296 | 57.870 | 69.444 81.019} 92.593) 104.167 
6/11.667| 23.333 | 35.000 | 46.667 | 58.333 | 70.000 81.667} 93.333) 105.000 
711.759] 23.519 | 35.278 | 47.0387 | 58.796 | 70.556 82.315} 94.074) 105.833 
8111. 852! 23.704 | 35.556 | 47.407 | 59.259 | 71.111 82.963) 94.815) 106.667 
-9/11.944| 23.889 | 35.833 | 47.778 | 59.722 | 71.667 83.611) 95.556) 107.500 
13.0/12.037| 24.074 | 36.111 | 48.148 | 60.185 | 72.222 84.259| 96.296) 108.333 
-1)12.130| 24.259 | 36.389 | 48.519 | 60.648 | 72.778 84.907} 97.037) 109.167 
-2\12.222| 24.444 | 36.667 | 48.889 | 61.111 | 73.333 85.556! 97.778) 110.000 
-3/12.315| 24.630 | 36.944 | 49.259 | 61.574 | 73.889 86.204) 98.519} 110.8383 
4/12.407| 24.815 | 37.222 | 49.630 | 62.0387 | 74.444 86.852; 99.259! 111.667 
512.500) 25.000 | 37.500 | 50.000 | 62.500 | 75.000 87.500) 100.000) 112.500 
612.593! 25.185 | 37.778 | 50.370 | 62.963 | 75.556 88.148} 100.741!) 113.333 
7112.685| 25.370 | 38.056 | 50.741 | 63.426 | 76.111 88.796) 101.481) 114.167 
© 8/12.778) 25.556 | 38.333 | 51.111 | 63.889 | 76.667 89.444] 102.222! 115.000 
9/12. 870) 25.741 | 38.611 | 51.481 | 64.352 | 77.222 90.093) 102.963) 115.833 
14.0/12.963) 25.926 | 38.889 | 51.852 | 64.815 | 77.778 90.741) 103.704) 116.667 
-1/13.056| 26.111 | 39.167 | 52.222 | 65.278 | 78.333 91.389] 104.444] 117.500 
.2/13.148] 26.296 | 39.444 | 52.593 | 65.741 | 78.889 92.037) 105.185) 118.333 
3113.241| 26.481 | 39.722 | 52.963 | 66.204 79.444 92.685} 105.926) 119.167 
-4/13.333| 26.667 | 40.000 | 58.333 | 66.667 | 80.000 93.333] 106.667) 120.000 
-5/13.426| 26.852 | 40.278 | 53.704 | 67.130 | 80.556 93.981} 107.407| 120.833 
-6113.519| 27.037-| 40.556 | 54.074 | 67.593 81.111 94.630) 108.148) 121.667 
713.611} 27.222 | 40.833 | 54.444 | 68.056 81.667 95.278! 108.889) 122.500 
8113.704| 27.407 | 41.111 | 54.815 | 68.519 | 82.222 95.926) 109.6380) 123.333 
-9/13.796| 27.593 | 41.389 | 55.185 | 68.981 | 82.778 96.574) 110.370) 124.167 
15.0/13.889) 27.778 | 41.667 | 55.556 ; 69.444 83.333 97.222) 111.111) 125.000 
-1/13.981| 277.963 | 41.944 | 55.926 | 69.907 | 83.889 97.870} 111.852) 125.833 
-2\14.074| 28.148 | 42.222 | 56.296 | 70.870 | 84.444 98.519) 112.593) 126.667 
3/14.167| 28.333 | 42.500 | 56.667 | 70.833 85.000 99.167) 113.383) 127.500 
A\14.259| 28.519 | -42.778 | 57.087 | 71.296 85.556 99.815] 114.074) 128.333 
.5/14.352! 28.704 | 48.056 | 57.407 | 71.759 86.111 | 100.463) 114.815) 129.167 
.6/14.444| 28.889 | 43.333 | 57.778 | 72.222 86.667 | 101.111) 115.556) 130.000 
-7|14.537| 29.074 | 43.611 | 58.148 | 72.685 87.222 | 101.759) 116.296) 180.8383 
-8114.630| 29.259 | 43.889 | 58.519 | 73.148 87.778 | 102.407) 117.037) 131.667 
9114.722| 29.444 | 44.167 | 58.889 | 73.611 88.333 | 103.056) 117.778) 132.500 
16.0/14.815| 29.630 | 44.444 | 59.259 | 74.074 88.889 } 108.704) 118.519) 133.3383 
-1/14.907| 29.815 | 44.722 | 59.630 | 74.537 89.444 | 104.352) 119.259) 134.167 
-2/15.000! 30.000 | 45.000 | 60.000 | 75.000 90.000 | 105.000) 120.000} 135.000 
-3/15.093| 30.185 | 45.278 | 60.370 | 75.463 90.556 | 105.648) 120.741) 135.833 
4115.185| 30.370 | 45.556 | 60.741 | 75.926 91.111 | 106.296) 121.481) 136.667 
.5115.278| 30.556 | 45.833 | 61.111 | 76.389 91.667 | 106.944) 122.222) 137.500 
-6115.370| 30.741 | 46.111 | 61.481 | 76.852 | 92.222 | 107.593) 122.963] 138.333 
-7115.463) 30.926 | 46.389 | 61.852 | 77.315 92.778 | 108.241) 123.704!) 139.167 
-8115.556] 31.111 | 46.667 | 62.222 | 77.778 93.333 | 108.889) 124.4441 140.000 
-9|15.648| 31.296 | 46.944 | 62.593 | 78.241 93.889 | 109.537) 125.185) 140.833 
17.0/15.'741| 31.481 | 47.222 | 62.963 | 78.704 94.444 | 110.185) 125.926} 141.667 
-1/15.833| 31.667 | 47.500 | 63.333 | 79.167 95.000 | 110.833) 126.667) 142.500 
-2/15.926, 31.852 | 47.778 | 68.704 | 79.630 95.556 | 111.481) 127.407) 143.333 
-3/16.019} 32.037 | 48.056 | 64.074 | 80.093 96.111 | 112.180} 128.148] 144.167 
4/16.111) 32.222 | 48.333 | 64.444 | 80.556 96.667 | 112.778) 128.889} 145.000 
-5116.204| 82.407 | 48.611 | 64.815-| 81.019 97.222 | 118.426) 129.630) 145.833 
6116. 296) 32.593 | 48.889 | 65.185 | 81.481 97.778 | 114.074) 180.370) 146.667 
-7116.389| 32.778 ,).49.167 | 65.556 81.944 | 98.333 | 114.722) 181.111) 147.500 
-8/16.481! 32.963 | 49.444 | 65.926 | 82.407 98.889 | 115.370) 181.852) 148.333 
9116.574| 33.148 | 49.722 | 66.296 | 82. 870 | 99.444 | 116.019} 132.593) 149.167 
18.0/16.667| 33.333 | 50.000 | 66.667 | 83. 333 100.000 | 116.667) 133.333) 150.000 








TABLE XXXI.—TRIANGULAR PRISMS 

































































WIDTH. 
2 3 4 5 6 7 8 9 
33.519 | 50.278 | 67.037 | 83.796) 100.556) 117.315] 134.074] 150.833 
33.704 | 50.556 | 67.407 | 84.259] 101.111) 117.963] 134.815] 151.667 
33.889 | 50.883 | 67.778 | 84.722) 101.667) 118.611] 135.556] 152.500 
34.074 | 51.111 | 68.148 | 85.185) 102.222) 119.259] 136.296] 153.333 
34.259 | 51.389 | 68.519 | 85.648) 102.778] 119.907| 137.037] 154.167 
34.444 | 51.667 | 68.889 | 86.111) 103.333] 120.556] 137.778] 155.000 
34.630 | 51.944 | 69.259 ) 86.574) 103.889} 121.204] 138.519) 155.833 
34.815 | 52.222 | 69.630 | 87.037) 104.444] 121.852) 139.259) 156.667 
35.000 | 52.500 | 70.000 |. 87.500) 105.000! 122.500) 140.000) 157.500 
35.185 | 52.778 | 70.370 | 87.963} 105.556} 123.148) 140.741! 158.333 
35.370 | 53.056 | 70.741 | 88.426) 106.111] 123.796] 141.481! 159.167 
35.556 | 53.333 | 71.111 | 88.889] 106.667) 124.444] 142.222) 160.000 
35.741 | 53.611 | 71.481 | 89.352) 107.222) 125.098) 142.963! 160.833 
35.926 | 53.889 | 71.852 | 89.815] 107.778) 125.741) 143.704| 161.667 
36.111 | 54.167 | 72.222 | 90.278) 108.333! 126.389] 144.444] 162.500 
36.296 | 54.444 | 72.593 | 90.741) 108.889] 127.037} 145.185] 163.333 
36.481 | 54.722 | 72.963 | 91.204) 109.444] 127.685) 145.926! 164.16 
36.667 | 55.000 | 73.333 | 91.667) 110.000) 128.333] 146.667! 165.00 
36.852 | 55.278 | 73.704 | 92.130) 110.556) 128.981] 147.407] 165.833 
37.037 | 55.556 | 74.074 | 92.593} 111.111! 129.630] 148.148] 166.667 
37.222 | 55.833 | 74.444 | 93.056} 111.667] 130.278] 148.889! 167.500 
87.407 | 56.111 | 74.815 | 93.519} 112.222) 130.926] 149.630! 168.333 
37.593 | 56.389 | 75.185 | 93.981) 112.778) 131.574) 150.370! 169.167 
37.778 | 56.667 | 75.556 | 94.444) 118.333) 132.222] 151.111! 170.000 
37.963 | 56.944 | 75.926 | 94.907) 113.889) 132.870] 151.852) 170.833 
38.148 | 57.222 | 76.296 | 95.370) 114.444] 133.519] 152.593] 171.667 
38.333 | 57.500 | 76.667 | 95.833) 115.000) 134.167] 153.333] 172.500 
38.519 | 57.778 | 77.037 | 96.296) 115.556] 134.815] 154.074). 173.333 
38.704 | 58.056 | 77.407 | 96.759) 116.111] 135.463] 154.815] 174.167 
38.889 | 58.8383 | 77.778 | 97.222] 116.667] 136.111] 155.556] 175.000 
39.074 | 58.611 ) 78.148 | 97.685! 117.222) 136.759! 156.296] 175.833 
39.259 | 58.889 | 78.519 | 98.148) 117.778) 137.407) 157.037! 176.667 
39.444 | 59.167 | 78.889 | 98.611} 118.333] 138.056) 157.778] 177.500 
39.630 | 59.444 | 79.259 | 99.074) 118.889] 138.704) 158.519] 178.332 
39.815 | 59.722 | 79.630 | 99.537) 119.444] 139.352] 159.259) 179.167 
40.000 | 60.000 | 80.000 | 100.000) 120.000} 140.000) 160.000) 180.000 
40.185 | 60.278 | 80.370 | 100.463) 120.556] 140.648) 160.741! 180.833 
40.370 | 60.556 | 80.741 | 100.926) 121.111] 141.296) 161.481] 181.667 
40.556 | 60.833 | 81.111 | 101.389} 121.667) 141.944) 162.222] 182.500 
40.741 | 61.111 | 81.481 | 101.852) 122.222] 142.593! 162.963] 183.333 
40.926 | 61.889 | 81.852 | 102.315; 122:778] 143.241] 163.704] 184.167 
41.111 | 61.667 | 82.222 | 102.778) 123.333] 143.889} 164.444] 185.000 
41.296 | 61.944 | 82.593 | 103.241) 123.889] 144.537] 165.185] 185.833 
41.481 | 62.222 | 82.963 | 108.704) 124.444) 145.185] 165.926] 186.667 
41.667 | 62.500 | 83.333 | 104.167; 125.000] 145.833] 166.667] 187.500 
41.852 | 62.778 | 83.704 | 104.630} 125.556] 146.481) 167.407! 188.333 
42.037 | 63.056 | 84.074 | 105.093) 126.111) 147.130] 168.148! 189.167 
42.222 | 63.333 | 84.444 | 105.556) 126.667) 147.778) 168.889] 190.000 - 
42.407 | 63.611 | 84.815 | 106.019} 127.222] 148.426! 169.630] 190.833 
42.593 | 63.889 | 85.185 | 106.481} 127.778] 149.074] 170.370] 191.667 
; , 42.778 | 64.167 | 85.556 | 106.944] 128.333] 149.722] 171.111] 192.500 
eleie 42.963 | 64.444 | 85.926 | 107.407; 128.889] 150.370! 171.852] 193.333 
, L 43.148 | 64.722 | 86.296 | 107.870} 129.444} 151.019] 172.593] 194.167 
i 43.333 | 65.000 | 86.667 | 108.333] 130.000! 151.667} 173.333] 195.000 
é 43.519 | 65.278 | 87.037 | 108.796] 130.556] 152.315] 174.074] 195.833 
3 ! 43.704 | 65.556 | 87.407 | 109.259] 131.111] 152.963) 174.815] 196.667 
5 : 43.889 | 65.883 | 87.778 | 109.722) 131.667] 158.611 175.556! 197.500 
ue 44.074 | 66.111 | 88.148 | 110.185] 132.222] 154.259] 176.296] 198.333 
t - 130) 44.259 | 66.389 | 88.519 | 110.648) 132.778] 154.907! 177.037] 199.167 
44.444 | 66.667 | 88.889 | 111.111] 183.333] 155.556] 177.778] 200.000 
eee! 


CU. YDS. PER 50 FT. 235 




















WIDTH. 
Es 

en 1 2 3 4 5 6 7 8 9 
~ 24.1/22.315) 44.630 | 66.944 | 89.259) 111.574) 133.889) 156.204! 178.519] 200.833 
.2/22.407| 44.815 | 67.222 | 89.630) 112.087) 184.444) 156.852] 179.259} 201.667 
.38/22.500} 45.000 | 67.500 | 90.000) 112.500) 135.000) 157.500) 180.000) 202.500 
-4)22.593| 45.185 | 67.778 | 90.370) 112.963) 185.556) 158.148) 180.741} 203.333 
.5/22.685} 45.370 | 68.056 | 90.741] 113.426) 136.111) 158.796} 181.481] 204.167 
.6/22.778) 45.556 | 68.383 | 91.111} 113.889] 136.667| 159.444) 182.222) 205.000 
.7/22.870| 45.741 | 68:611 | 91.481) 114.352} 187.222) 160.093) 182.963] 205.833 
.8/22.963) 45.926 | 68.889 | 91.852) 114.815) 187.778) 160.741] 183.704] 206.667 
.9/23.056) 46.111 | 69.167 | 92.222) 115.278] 188.333) 161.389] 184.444) 207.500 
25.0/23.148) 46.296 | 69.444 | 92.593) 115.741] 188.889] 162.087) 185.185) 208.333 
23.241} 46.481 | 69.722 | 92.963} 116.204) 189.444) 162.685) 185.926) 209.167 





23.333} 46.667 | 70.000 | 93.333} 116.667) 140.000; 163.333) 186.667) 210.000 
23.426) 46.852 | 70.278 | 93.704) 117.130} 140.556) 163.981) 187.407) 210.833 
23.519] 47.037 | 70.556 | 94.074) 117.593) 141.111] 164.630} 188.148) 211.667 
28.611) 47.222 | 70.833 | 94.444) 118.056} 141.667) 165.278} 188.889! 212.500 
23.704) 47.407 | 71.111 | 94.815} 118.519] 142.222) 165.926] 189.630} 213.333 
23.796) 47.593 | 71.389 | 95.185} 118.981} 142.778] 166.574} 190.370) 214.167 
23.889) 47.778 | 71.667 | 95.556; 119.444) 148.333) 167.222) 191.111) 215.000 
23.981) “47.963 | 71.944 | 95.926} 119.907) 143.889} 167.870) 191.852) 215.833 
24.074} 48.148 | 72.222 | 96.296) 120.370} 144.444) 168.519} 192.593) 216.667 


24.167} 48.3383 | 72.500 | 96.667} 120.833) 145.000) 169.167) 193.333) 217.500 
24.259) 48.519 | 72.778 | 97.0 7) 121.296) 145.556) 169.815) 194.074) 218.333 
24.352) 48.704 | 73.056 | 97.407) 121.759) 146.111) 170.463) 194.815) 219.167 
24.444! 48.889 | 73.333 | 97.778} 122.222} 146.667] 171.111] 195.556) 220.000 
24.587) 49.074 | 73.611 | 98.148) 122.685) 147.222) 171.759) 196.296) 220.833 
24.630). 49.259 | 73.889 | 98.519} 123.148) 147.778) 172.407] 197.037) 221.667 
24.722) 49.444 | 74.167 | 98.889} 123.611) 148.333) 173.056) 197.778] 222.500 
24.815| 49.630 | 74.444 | 99.259} 124.074; 148.889) 173.704} 198.519) 223.333 
24.907| 49.815 | 74.722 | 99.630] 124.537} 149.444] 174.352} 199.259) 224.167 
25.000) 50.000 | 75.000 | 100.000; 125.000) 150.000) 175.000) 200.000) 225.000 


25.093} 50.185 | 75.278 | 100.370) 125.463) 150.556) 175.648} 200.741) 225.833 
25.185} 50.370 | 75.556 | 100.741) 125.926} 151.111) 176.296) 201.481} 226.667 
25.278} 50.556 | 75.833 | 101.111) 126.389) 151.667) 176.944! 202.222) 227.500 
25.370! 50.741 | 76.111 | 101.481) 126.852) 152.222) 177.593) 202.963) 228.333 
25.463) 50.926 | 76.389 | 101.852) 127.315) 152.778) 178.241| 203.704) 229.167 
25.556} 51.111 | 76.667 | 102.222| 127.778) 153.333) 178.889) 204.444} 230.000 
25.648) 51.296 | 76.944 | 102.593) 128.241) 153.889) 179.537) 205.185) 230.833 
25.741| 51.481 | 77.222 | 102.963} 128.704) 154.444) 180.185] 205.926) 231.667 
25.833) 51.667 | 77.500 | 103.333) 129.167) 155.000) 180.833) 206.667) 232.500 
25.926) 51.852 | 77.778 | 103.704) 129.680) 155.556; 181.481} 207.407} 233.333 


26.019) 52.037 | 78.056 | 104.074) 130.093} 156.111} 182 130} 208.148) 234.167 
(26.111) 52.222 | 78.833 | 104.444) 130.556) 156.667| 182.778) 208.889) 235.000 
26.204) 52.407 | 78.611 | 104.815) 131.019) 157.222) 183.426] 209.630} 235.833 
26.296) 52.593 | 78.889 | 105.185) 131.481) 157.778) 184.074) 210.370) 236.667 
26.389) 52.778 | 79.167 | 105.556) 181.944) 158.333) 184.722) 211.111] 237.500 
26.481} 52.963 | 79.444 | 105.926) 182.407; 158.889) 185.370} 211.852) 238.333 
26.574) 53.148 | 79.722 | 106.296) 132.870; 159.444) 186.019) 212.593) 239.167 
26.667; 53.333 | 80.000 | 106.667) 133.333) 160.000) 186.667} 213.333) 240.000 
26.759) 53.519 | 80.278 | 107.087) 133.796) 160.556) 187.315} 214.074) 240.833 
26.852) 53.704 | 80.556 | 107.407) 134.259) 161.111} 187.963) 214.815) 241.667 


26.944) 53.889 | 80.833 | 107.778) 134.722} 161.667) 188.611) 215.556) 242.500 
(27.037) 54.074 | 81.111 | 108 148) 135.185) 162.222} 189.259) 216.296) 243.333 
: nee 135.648] 162.778) 189.907) 217.037) 244.167 
136.111} 163.333} 190.556) 217.778) 245.000 
-259} 136.574) 163.889) 191.204) 218.519) 245.833 
27.407) 54.815 | 82.222 | 109.630) 137.037] 164.444] 191.852] 219.259] 246.667 
: 137.500) 165.000} 192.500} 220.000) 247.500 

137.963) 165.556) 193.148} 220.741) 248.333 
188.426) 166.111) 193.796} 221.481) 249.167 
30. 027. 778| 55.556 | 83.333 | 111.111] 138.889] 166.667| 194.444] 222.222] 250.000 








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236 TABLE XXXI.—TRIANGULAR PRISMS 
WIDTH. 
6 
im} 4 2 3 4 5 6 7 8 
30.1/27.870| 55.741 | 88.611] 111.481] 139.352] 167.222] 195.093] 222.963 
"2/27.963| 55.926 | 83.889) 111.852] 139.815] 167.778] 195.741/223.704 
"3/28.056| 56.111 | 84.167] 112.222) 140.278] 168.333| 196.389] 224.444 
‘4/28. 148] 56.296 | 84.444) 112.593) 140.741] 168.889] 197.037] 225.185 
-5/28.241| 56.481 | 84.722; 112.963) 141.204] 169.444] 197.685! 225.926 
6128. 333| 56.667 | 85.000] 113.333] 141.667! 170.000| 198.333) 226.667 
“7/28. 426) 56.852 | 85.278] 113.704! 142.130) 170.556| 198.981| 227.407 
"8128.519| 57.037 | 85.556) 114.074] 142.593] 171.111| 199.630| 228.148 
'9/28.611| 57.222 ; 85.833| 114.444) 143.056] 171.667] 200.278! 228.889 
31.0/28.704| 57.407 | 86.111] 114.815) 143.519] 172.222] 200.926] 229.630 
.1128.796| 57.593 | 86.389) 115.185] 143.981] 172.778] 201.574] 230.370 
"2/28. 889| 57.778 | 86.667| 115.556) 144.444] 173-333] 202.222) 231.111 
.38/28.981] 57.963 86.944! 115.926} 144.907] 173.889} 202.870) 231.852 
‘4129.074| 58.148 | 87.222] 116.296) 145.370] 174.444] 203.519] 232.593 
'5|29.167| 58.333 | 87.500) 116.667] 145.833! 175.000! 204.167| 233.333 
629.259] 58.519 | 87.778] 117.037| 146.296] 175.556] 204.815| 234.074 
"7/29 .352| 58.704 | 88.056] 117.407] 146.759] 176.111] 205.463! 234.815 
"8129. 444] 58.889 | 88.333’ 117.778] 147.222] 176.667! 206.111] 235.556 
"9 29.537| 59.074 | 88.611] 118.148] 147.685] 177.222] 206.759] 936.296 
32.0/29.630| 59.259 | 88.889] 118.519) 148.148] 177.778] 207.407| 237.037 
1129.722| 59.444 | 89.167) 118.889] 148.611! 178.333] 208.056] 237.778 
-2'29.815| 59.630 | 89.444) 119.259] 149.074] 178.889] 208.704] 238.519 
°329.907| 59.815 | 89.722| 119.630] 149.587| 179.444] 209.352] 239.259 
"4/30.000| 60.000 | 90.000] 120.000] 150.000/ 180.000; 210.000| 240.000 
"5/30.093| 60.185 | 90.278] 120.370] 150.463] 180.556| 210.648] 240.741 
"630.185! 60.370 | 90.556] 120.741) 150.926] 181.111/ 211.296) 241.481 
730.278] 60.556 | 90.833] 121.111] 151.389) 181.667) 211.944] 242.292 
"g'30.370| 60.741 | 91.111) 121.481| 151.852] 182.222) 212.593] 242.963 
"9 30.463] 60.926 | 91.389] 121.852] 152.315) 182.778] 213.241] 243.704 
33.0,30.556] 61.111 | 91.667| 122.222] 152.778] 183.333] 213.889] 244.444 
130.648] 61.296 | 91.944) 122.593! 158.241! 183.889] 214.537) 245.185 
'2/30.741| 61.481 | 92.222] 122.963 153.704] 184.444] 215.185| 245.926 
"330.8331 61.667 | 92.500! 123.333] 154.167) 185.000) 215.833] 246. 667 
4'30.926| 61.852 | 92.778] 123.704] 154.630! 185.556) 216.481| 247.407 
'5131.019| 62.037 | 93.056] 124.074] 155.093) 186.111) 217.130] 248.148 
631.111] 62.222 | 93.333] 124.444] 155.556] 186.667| 217.7781 248.889 
731.204] 62.407 | 93.611) 124.815] 156.019) 187.222] 218.426] 249.630 
"8/31.296| 62.593 | 93.889) 125.185] 156.481| 187.778) 219.074| 250.370 
"9'31.389| 62.778 | 94.167) 125.556 156.944) 188.333) 219.722) 251.111 
34.031.481| 62.963 | 94.444] 125.926] 157.407 188.889] 220.370) 251.852 
.1/31.574] 63.148 | 94.722] 126.296] 157.870] 189.444) 221.019] 252.593 
'231.667| 63.333 | 95.000] 126.667| 158.333] 190.000 221.667] 253.333 
'331.759| 63.519 | 95.278! 127.037| 158.796] 190.556) 222.315) 254.074 
‘4'31.852] 63.704 | 95.556) 127.407] 159.259] 191.111) 222.963] 254.815 
'5/31.944| 63.889 | 95.833) 127.778] 159.722] 191.667) 223.611] 255.556 
-6.32.037| 64.074 | 96.111) 128.148] 160.185] 192.222] 224.259] 256 296 
“7/32.130| 64.259 | 96.389] 128.519] 160.648] 192.778] 224.907| 257.037 
"8/32. 222| 64.444 | 96.667] 128.889] 161.111| 193.333) 225.556] 257.778 
'9/32.315| 64.630 | 96.944] 129.259] 161.574) 193.889) 226.204] 258.519 
35.0/32.407| 64.815 | 97.222] 129.630) 162.037} 194.444) 226.852) 259.259 
132.500] 65.000 | 97.500! 130.000/ 162.500! 195.000) 227.500! 260.000 
"2/32 593] 65.185 | 97.778] 130.370] 162.963! 195.556 228.148] 260.741 
'3/32.685| 65.370 | 98.056] 130.741| 163.426] 196.111' 228.796 261.481 
'4132.778| 65.556 | 98.333] 131.111| 163.889] 196.667] 229.444] 262.299 
'5/32.870| 65.741 | 98.611] 131.481! 164.352] 197.222] 230.093] 262.963 
-6/32.963| 65.926 | 98.889] 131.852] 164.815] 197.778] 230.741] 263.704 
-7|33.056| 66.111 | 99.167] 132.222) 165.278] 198.333] 231.389] 264.444 
-8133.148] 66.296 | 99.444| 132.593! 165.741! 198.889] 232.037] 265.185 
-9/33.241| 66.481 | 99.722] 132.963) 166.204! 199.4441 232.685! 265.926 
36.0'33.333| 66.667 | 100.000, 133.333) 166.667} 200.000] 233.333) 266.667 























9 





250.833 _ 
251.667 
252.500 
253.333 
204.167 
255 . 000 
295. 833 
256. 667 
297.500 
258.333 


259. 167 
260.000 
260.833 
261.667 
262.500 
263. 333 
264.167 
265.000 
265 . 833 
266. 667 


267.500 
268 . 333 
269. 167 
270.000 
270. 833 
271.667 
212.500 
273.333 
274.167 
275.000 


275.833 
276. 667 
277.500. 
278.333 
279.167 
289.000 
280. 833 
281.667 
282.500 
283.333 


284.167 
285.000 
285. 833 
286. 667 
287.500 
288.333 
289.167 
290.000 
290. 833 
291. 667 


292.500 
293.333 
294.167 
295.000 
295. 833 
296. 667 
297.500 
298 . 333 
299.167 
300. 000 





CU. YDS. PER 50 FT. 237 





Nyc bi OY Rs & 





3 4 5 6 7 8 9 


33.426) 66.852 | 100.278] 133.704| 167.130) 200.556) 233.981) 267.407) 300.833 
33.519) 67.037 | 100.556) 134.074) 167.593) 201.111) 234.630] 268.148) 301.667 
33.611} 67.222 | 100.833} 134.444) 168.056) 201.667| 235.278) 268.889) 302.500 
33.704) 67.407 | 101.111) 134.815] 168.519] 202.222) 235.926) 269.630} 303.333 
33.796) 67.593 | 101.389) 135.185) 168.981} 202.778) 236.574) 270.370] 304.167 
33.889} 67.778 | 101.667) 135.556) 169.444) 203.333) 237.222) 271.111) 305.000 
33.981! 67.963 | 101-944] 135.926) 169.907) 203.889) 237.870) 271.852) 305.833 
34.074) 68.148 | 102.222) 136.296) 170.370} 204.444) 238.519) 272.593) 306.667 
34.167) 68.333 | 102.500) 136.667; 170.833) 205.000} 239.167) 273.333) 307.500 
34.259) 68.519 | 102.778) 137.037) 171.296} 205.556) 239.815| 274.074) 308.333 


34.352| 68.704 | 103.056} 137.407) 171.759} 206.111] 240.463) 274.815) 309.167 
34.444| 68.889 | 103.333) 187.778) 172.222} 206.667) 241.111} 275.556] 310.000 
34.537| 69.074 | 103.611) 188.148) 172.685) 207.222) 241.759) 276.296) 310.833 
34.630} 69.259 | 103.889) 138.519) 173.148) 207.778) 242.407) 277.037) 311.667 
34.722) 69.444 | 104.167] 188.889} 173.611] 208.333] 248.056) 277.778) 312.500 
34.815} 69.630 | 104.444) 139.259) 174.074, 208.889) 248.704; 278.519) 313.333 
34.907) 69.815 | 104.722} 139.630) 174.537) 209.444} 244.352) 279.259] 314.167 
35.000) 70.000 | 105.000} 140.000} 175.000} 210.000} 245.000) 280.000) 315.000 
35.093) 70.185 | 105.278) 140.370) 175.463) 210.556| 245.648] 280.741) 315.833 
35.185} 70.370 | 105.556) 140.741) 175.926; 211.111) 246 296) 281 481] 316.667 


35.278} 70.556 | 105.833} 141.111) 176.389) 211.667) 246.944) 282.222) 317.500 
35.370} 70.741 | 106.111} 141.481] 176.852) 212.222) 247.5938) 282.963) 318.333 
35.463) 70.926 | 106.389} 141.852) 177.315} 212.778) 248.241] 283 704) 319.167 
35.556) 71.111 | 106.667) 142.222) 177.778) 213.333) 248.889] 284.444) 320.000 
35.648) 71.296 | 106.944) 142.593) 178.241] 213.889) 249.537) 285.185) 320.833 
35.741| 71.481 | 107.222) 142.963) 178.704) 214.444) 250.185] 285.926) 321.667 
35.833) 71.667 | 107.500} 143.383) 179.167; 215.000} 250.833) 286.667) 322.500 
35.926| 71.852 | 107.778) 143.704) 179.630) 215.556} 251.481) 287.407) 323.333 
36.019) 72.037 | 108.056} 144.074) 180.093) 216.111) 252.130) 288.148) 324.167 
36.111) 72.222 | 108.333} 144.444) 180.556| 216.667} 252.778) 288.889) 325.000 


: : 108.611} 144.815} 181.019) 217.222) 253.426) 289.630) 325.833 
36.296) 72.593 | 108.889} 145.185) 181.481] 217.778) 254.074| 290.370) 326.667 
36.389) 72.778 | 109.167} 145.556) 181.944) 218.333) 254.722) 291.111] 327.500 
36.481) 72.963 | 109.444) 145.926) 182.407) 218.889) 255.370| 291.852] 328.333 
36.574! 73.148 | 109.722} 146.296) 182.870) 219.444) 256.019} 292.593] 329.167 
36.667) 73.333 | 110.000) 146.667) 183.333) 220.000) 256.667) 293.333) 330.000 
36.759| 78.519 | 110.278] 147.0387| 183.796} 220.556) 257.315] 294.074| 330.833 
36.852) 73.704 | 110.556) 147.407) 184.259} 221.111) 257.963) 294.815) 331.667 
36.944) 73.889 | 110.833] 147.778) 184.722] 221.667) 258.611} 295.556) 332.500 
37.037| 74.074 | 111.111} 148.148) 185.185} 222.222) 259.259] 296.296] 333.333 


37.130} 74.259 | 111.389) 148.519) 185.648} 222.778) 259.907) 297.037] 334.167 
37.222| 74.444 | 111.667| 148.889|.186.111] 223.333) 260.556} 297.778] 335.000 
37.315; 74.630 , 111.944) 149.259) 186.574| 223.889) 261.204) 298.519) 335.833 
37.407| 74.815 | 112.222) 149.630} 187.037] 224.444] 261.852) 299.259] 336.667 
37.500} 75.000 | 112.500} 150.000} 187.500} 225.000; 262.500) 300.000} 337.500 
37.593] ‘75.185 | 112.778) 150.370} 187.963} 225.556; 263.148} 300.741} 338.333 
37.685) 75.370 | 113.056) 150.741} 188.426} 226.111} 263.796} 301.481) 339.167 
37.778] 75.556 | 113.333] 151.111] 188.889} 226.667) 264.444) 302.222] 340.000 
37.870} 75.741 | 113.611) 151.481) 189.352) 227.222} 265.093) 302.963} 340.833 
37.963] 75.926 | 113.889} 151.852} 189.815} 227.778) 265.741] 303.704) 341.667 


38.056) 76.111 | 114.167} 152.222) 190.278) 228.333) 266.389) 304.444) 342.500 
38.148} 76.296 | 114.444) 152.593) 190.741) 228.889) 267.087) 305.185) 343.333 
38.241) 76.481 | 114.722) 152.963) 191.204) 229.444! 267.685] 305.926) 244.167 
38.333} 76.667 | 115.000} 153.333) 191.667) 230.000) 268.333) 306.667! 345.000 
38.426} 76.852 | 115.278) 153.704]. 192.130} 230.556) 268.981) 307.407) 345.833 
38.519] 77.037 | 115.556) 154.074) 192.593) 231.111) 269.630} 308.148) 346.667 
38.611} 77.222 | 115.833) 154.444) 193.056) 231.667) 270.278) 308.889] 347.500 
38.704) 77.407 | 116.111) 154.815) 193.519) 232.222) 270.926) 309.630) 348.333 
38.796] 77.593 | 116.389) 155.185) 193.981) 232.778) 271.574) 310.370) 349.167 
[38.889 77.778 | 116.667) 155.556) 194.444) 233.333) 272.222) 311.111/ 350.000 


| Het. 
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238 


TABLE XXXI.—TRIANGULAR PRISMS 








| Het. 
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44. 


WOH DOWODARNERWHH CODIROTRwWHOH COBYISMAWNE 
= sD GOS 
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> 
en 
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= 
a 





= 
[oo] 














3 





116.944 
117.222 
117.500 
117.778 
118.056 
118.333 
118.611 
118.889 
119.167 
119.444 


119.722 
120.000 
120.278 
120.556 
120.833 
121.111 
121.389 
121.667 
121.944 
122.222 


122.500 
122.778 
123. 056 
123.333 
123.611 
123.889 
124. 167 
124.444 
124.722 
125.000 


125.278 
125.556 
125.833 
126.111 
126.389 
126. 667 
126.944 
127. 222 
127.500 
127.778 


128. 056 
128 . 333 
128.611 
128.889 
129.167 
129. 444 
129.722 
130.000 
130.278 
130.556 


130. 833 
131.111 
131.389 
131.667 
131.944 
132.222 
132.500 
132.778 
133.056 
133.333 





4 





155.926 
156.296 
156.667 
157.037 
157.407 
157.778 
158.148 
158.519 
158. 889 
159.259 


159. 630 
160.000 
160.370 
160.741 
161.111 
161.481 
161.852 
162.222 
162.593 
162.963 


163.333 
163.704 
164.074 
164.444 
164.815 
165.185 
165.556 
165.926 
166.296 
166. 667 


167.037 
167.407 
167.778 
168.148 
168.519 
168. 889 
169.259 
169. 630 
170.000 
170.370 


170.741 
171.111 
171.481 
171.852 
172.222 
172.593 
172.963 
173.333 
173.704 
174. 074 


174.444 
174.815 
175.185 
175.556 
175.926 
176.296 
176. 667 
177.037 
177.407 
177.778 








WIDTH. 





5 





194.907 
195.370 
195.833 
196.296 
196.759 
197.222 
197.685 
198.148 
198.611 
199.074 


199.537 
200.000 
200.463 
200.926 
201.389 
201.852 
202.315 
202.778 
203.241 
203.704 


204.167 


208.333 


208.796 
209. 259 
209.722 
210.185 
210.648 
211.111 
211.574 
212.037 
212.500 
212.963 


218.426 
213.889 
214.352 
214.815 
215.278 
215.741 
216.204 
216.667 
217.130 
217.593 


218.056 


218.519; 


218.981 
219.444 
219.907 
220.370 
220. 833 
221.296 
221.759 
222.002 








6 





233.889 
284.444 
235.000 
235.556 
236.111 
236.667 
237.222 
237.778 
238.333 
238. 889 


239.444 
240.000 
240.556 
241.111 
241.667 
242.222 
242.778 
243.333 
243.889 
244.444 


245.000 


249.444 
250.000 


250.556 
201.111 
251.667 
252.228 
252.778 
203.333 
258. 889 
254.444 
255.000 
209.556 


256.111 
256.667 
207. dee 
207.778 
258 . 833 
258. 889 
259.444 
260.000 
260.556 
261.111 


261.667 
262.228 
262.778 
263.333 
263 . 889 
264.444 
265.000 
265.556 
266.111 
266. 667 











7 





272.870 
273.519 
274.167 
274.815 
275.463 
276.111 
276.759 
277.407 
278.056 
278.704 


279.352 
280.000 
280.648 
281.296 
281.944 
282.593 
283.241 
283 . 889 
284.537 
285.185 


285 . 833 
286.481 
287.130 
287.778 
288. 426 
289.074 
289.722 
290.370 
291.019 
291.667 


292.315 
292.963 
293.611 
294. 259 
294.907 
295.556 
296.204 
296 . 852 
297.500 
298.148 


298.796 
299.444 
300.093 
300.741 
301.389 
302.037 
302.685 
303.333 
303.981 
304. 630 


305.278 
305. 926 
306.574 
307.222 
307.870 
308.519 
309.167 
309.815 
310.463 
311.111 








> 


8 





311.852 
312.593 
313.333 
314.074 
314.815 
315.556 
316.296 
317.037 
317.778 
318.519 


319.259 
320.000 
320.741 
321.481 
322.222 
822.963 
323.704 
324.444 
825.185 
325.926 


826. 667 
327.407 
328.148 
328. 889 
329. 630 
330.370 
331.111 
331.852 
332.593 
333.333 


334.074 
334.815 


340.741 


341.481 
342.222 
342.963 
343.704 
344.444 
345.185 
345.926 
346.667 
347. 407 
348.148 


348. 889 
349 . 630 


350. 370) 


351.111 
351.852 
352.593 
353. 333 
354.074 
354. 815 
355.556 














9 





350. 833 
351. 667 
352.500 
353 . 333 
354. 167 
355. 000 
355 . 833 
356 . 667 
357.500 
358 . 333 


359. 167 
360.000 


360. 833 _ 


361. 667 
362.500 
363.333 
364.167 
365.000 
365. 833 
366. 667 


367.500 
368. 333 
369. 167 
370.000 
370.833 
371.667 
372.500 
373.333 
374.167 
375.000 


375. 833 


376.667 


383.333 
384.167 


391.667 


392.500 
393.333 
394. 167 
395.000 
395. 833 
396. 667 
397.500 
398.333 
399. 167 
400.000 





CU. YDS. PER 50 FT. 


239 





| Het. 


m™ 
co 


ee 
© 


ou 
So 


oO 
= 


on 
nw 


or 
oo 


= 


SOCOM ENR WDE SODOMIRTNRWNH CODARTNEWDWH CHODIRTMAWHH COMORIBTNRWHOH CBDIRONRwWoOHe 





ESE EE 
— OOo 0co 
Besse 


PS 


5.278 


i 

BHC 
wn 
Ry 
or) 


4 
4 
45 . 833 
4 
4 
4 





49.907 





50.000 








96.481 
96.667 
96. 852 
97.037 
97.222 
97.407 


100.000 





| 139.167 





3 





133.611 
183.889 
134. 167 
134.444 
134.722 
135.000 
135.278 
135.556 
135.833 
136.111 


136.389 
136.667 
136.944 
187.222 
137.500 
137.778 
188.056 
138.333 
138.611 
138.889 


139.444 
139.722 
140.000 
140.278 
140.556 
140.833 
141.111 


4 


WIDTH. 


5 


6 


vs 








141.389 
141.667 


141.944 


| 142.222 


142.500 
142.778 
143.056 
143.333 
143.611 
143.889 
144.167 
144.444 


144.722 
145.000 
145.278 
145.556 
145. 833 
146.111 
146.389 
146. 667 
146.944 
147.222 


147.500 
147.778 
148 056 
148.333 
148 611 
148.889 
149.167 
149-444 
149.722 
150.000 





178.148 
178.519 
178. 889 
179. 259 
179.630 
180.000 
180.370 
180.741 
181.111 
181.481 


181. 852 
182.222 
182.593 
182.963 
183. 333 
183.704 
184. 074 
184.444 
184.815 
185.185 


185.556 
185.926 
186.296 
186. 667 
187.037 
187.407 
187.778 
188. 148 
188.519 
188.889 


189.259 
189.630 
190. 000 
190.370 
190.741 
191.111 
191.481 
191.852 
192.222 
192.593 


192.963 
193.333 
193.704 
194.074 
194.444 
194.815 
195.185 
195.556 
195.926 
196.296 


196. 667 
197.037 
197.407 
197.778 
198.148 
198.519 
198.889 
199. 259 
199. 630 
200.000 





222.685 
223.148 
223.611 
224.074 
224.537 
225.000 
225.463 
225.926 
226.389 
226.852 


227.315 
227.778 
228.241 
228.704 
229.167 
229.630 
230. 093 
230.556 
231.019 
231.481 


231.944 
232.407 
232.870 
233.333 
233.796 
234.259 
234.722 
230.185 
235.648 


236.111). 


236.574 
237 .037 
237.500 
237 . 963 
238.426 
238 . 889 
239.352 
239.815 
240.278 
240.741 


241. 204 
241. 667 
242.130 
242.593 
243056 
243 519 
243.981 
244 444 
244.907 
245. 370 


245 . 833 
246.296 
246.759 
247.222 
247.685 
248. 148 
248 611 
249.074 
249.537 
250.000 








267.222 
267.778 
268 333 
269. 889 
269.444 
270.000 
270.556 
e71.111 
271.667 
272.222 


272.778 
273.333 
2783 . 889 
274.444 
275.000 
275.556 
276.111 
276. 667 
el 222 
277.778 


278.333 


283 . 889 
284.444 
285.000 
285.556 
286.111 
286. 667 
287 . 222 
287.778) 
288 . 833 
288. 889 


289.444 
290.000 
290.556 
291.111 
291. 667 
292.222 
292.778 
293. 333 
293.889 
294.444 


295.000 
295.556 
296.111 
296. 667 
297 . 222 
297.778 
298. 333 
298 . 889 
299.444 
300.000 





311.759 
312.407 
313.056 
313.704 
314.352 
315.000 
315.648 
316.296 
316.944 
317.593 


318.241 
318.889 
319.537 
320.185 
320.833 
821.481 
822.130 
822.778 
823. 426 
824.074 


324.722 
325 .370 
826.019 
326.667 
827.315 
327.963 
328.611 
829.259 
329.907 
330.556 


331.204 
331.852 
832.500 
333.148 
333.796 
334.444 
335. 093 
335.741 
336.389 
337.037 


337 . 685 
338 3833 
338.981 
339. 630 
340.278 
340.926 
341.574 
342.222 
342.870 
343.519 


344.167 
344.815 
345. 463 
346.111 
346.759 
347 407 
348.056 
348.704 
349 352 
350.000 








356.296 
357. 087 
357.778 
358.519 
359. 259 
360.000 
360.741 
361.481 
852.222 
362 . 963 


363.704 
364.444 
365.185 
365.926 
366. 667 
367. 407 
368. 148 
368 . 889 
369 . 630 
370.370 


371.111 
871.852 
372.593 
373 . 333 
374 074 
374.815 
375 .556 
376.296 
3877 . 037 
377.778 


378.519 
879.259 
380.000 
380.741 
381.481 
882.222 
382.963 
383. 704 
384.444 
385 .185 


385 . 926 
386 . 667 
387.407 
388 . 148 
388. 889 
389 . 630 
390.370 
391.111 
391. 852 
392.593 


393 .333 
394.074 
394.815 
395.556 
396. 296 
397 .037 
397.778 
398 .519 
399. 259 
400.000 








400. 833 
401.667 
402.500 
403 . 333 
404.167 
405 . 000 
405 . 833 
406.667 
407.500 
408 . 333 


409.167 
410.000 


416.667 


417.500 
418.333 
419.167 
420.000 
420.833 
421.667 
422.500 
423.333 
424.167 
425.000 


425. 833 
426.667 
427.500 
428 .333 
429.167 
430.000 
430.833 
431. 667 
432.500 
433 .333 


434.167 
435. 000 
435 . 833 
436 . 667 
437.500 
438 . 333 
439.167 
440.000 
440. 833 
441 667 


442.500 
443 333 
444.167 
445.000 
445 833 
446 . 667 
447.500 
448 333 
449.167 
450.000 





240 TABLE XXXI.—TRIANGULAR PRISMS 
CU. YDS. PER 50 FT. 























WIDTH. 
ti 
aa] 1 2 3 4 5 6 ote ,8 
54.1/50.093) 100.185) 150.278) 200.370} 250.463) 300.556] 350.648] 400.741 


50.185} 100.370; 150.556} 200.741} 250.926) 301.111} 351.296] 401.481 
50.278) 100.556/ 150.833} 201.111) 251.389} 301.667] 351.944) 402.222 
50.370) 100.741) 151.111; 201.481} 251.852] 302.222] 352.593) 402.963 
50.463) 100.926) 151.389} 201.852] 252.315} 302.778] 353.241) 403.704 
50.556; 101.111) 151.667} 202.222) 252.778) 303.333] 353.889} 404.444 
50.648) 101.296) 151.944) 202.598] 253.241} 303.889} 354.537| 405.185 
101.481} 152.222) 202.963] 253.704] 304.444] 355.185] 405.926 
50.833] 101.667] 152.500) 203.333] 254.167 305.000} 355.833) 406.667 
50.926} 101.852) 152.778) 203.704] 254.630) 305.556] 356.481] 407.407 


51.019) 102.037) 153.056} 204.074} 255.093] 306.111] 357.130) 408.148 
51.111) 102.222) 153.333) 204.444! 255.556) 306.667] 357.778) 408.889 
51.204) 102.407) 153.611] 204.815) 256.019) 307.222] 358.426 409.630 
51.296) 102.593) 153.889| 205.185) 256.481) 307.778] 359.074 410.370 
102.778} 154.167) 205.556] 256.944) 308.333] 359.722) 411.111 
102.968} 154.444) 205.926 257.407) 308.889; 360.370] 411.852 
51.574] 103.148] 154.722] 206.296 297.870) 309.444! 361.019} 412.593 
51 667) 103.333) 155.000} 206.667; 258.333) 310.000} 361.667) 413.333 
51.759} 103.519} 155.278] 207.037| 258.796) 310.556} 362.315} 414.074 
51.852} 103.704) 155.556} 207.407) 259.259] 311.111] 362.963) 414.815 


51.944) 103.889} 155.833} 207.778) 259.722} 311.667) 363.611) 415.556 
52.037! 104.074) 156.111} 208.148} 260.185] 312.222) 364.259] 416.296 
52.180) 104.259} 156.389} 208.519! 260.648) 312.778] 364.907| 417.037 
52.222) 104.444) 156.667} 208.889] 261.111) 313.333] 365.556) 417.778 
52.315) 104.680) 156.944} 209.259] 261.574) 313.889) «366.204 418.519 
52.407} 104.815) 157.222} 209.630) 252.037) 314.444) 366.852] 419.259 
52.500} 105.000) 157.560) 210.000) 262.500) 315.000) 367.500| 420.000 
52.593) 105.185) 157.778) 210.370} 262.963) 315.556] 368.148) 420.741 
52.685} 105.370) 158.056) 210.741} 263.426) 316.111] 368.796) 421.481 
52.778} 105.556) 158.333} 211.111) 263.889) 316.667) 369.444) 422.222 


52.870} 105.741) 158.611] 211.481) 264.352) 317.222) 370.093) 422.963 
52.963) 105.926) 158.889) 211.852) 264.815) 317.778] 370.741] 423.704 
53.056} 106.111) 159.167} 212.222) 265.278] 318.333] 371.389) 424.444 
53.148) 106.296) 159.444] 212.593) 265.741] 318.889] 372.037) 425.185 
53.241) 106.481) 159.722) 212.963] 266.204) 319.444! 372.685] 425.926 
53.333) 106.667) 160.000) 213.333) 266.667) 320.000) 373.333] 426.667 
53.426) 106.852) 160.278) 213.704) 267.130) 320.556) 373.981] 427.407 
53.519) 107.037) 160.556) 214.074| 267.593) 321.111) 374.630) 428.148 
107.222) 160.833) 214.444) 268.056) 321.667) 375.278) 428.889 
53.704| 107.407) 161.111 214.815) 268.519) 322.222) 375.926) 429.630 


53.796) 107.593) 161.389) 215.185) 268.981) 322.778] 376.574! 430.370 
53.889| 107.778) 161.667) 215.556) 269.444) 323.333] 377.222] 431.111 
53.981) 107.963) 161.944) 215.926) 269.907) 323.889] 377.870) 431.852 
54.074) 108.148) 162.222) 216.296) 270.370] 324.444] 378.519] 432.593 
54.167| 108.333) 162.500} 216.667] 270.833] 325.000) 379.167) 433.333 
54.259} 108.519) 162.778) 217.037) 271.296] 325.556] 379.815] 434.074 
54.352} 108.704) 163.056) 217.407) 271.1759} 326.111] 380.463) 434.815 
54.444) 108.889) 163.333] 217.778] 272.222] 326.667] 381.111) 435.556 
54.537; 109.074) 163.611] 218.148 ste 327.222| 381.759} 436.296 
54.630} 109.259) 163.889) 218.519) 278.148] 327.778] 382 407| 437.037 


54.722) 109.444) 164.167] 218.889} 273.611] 328.333] 383.056] 437.778 
54.815) 109.630) 164.444) 219.259] 274.074] 328.889] 3883.704| 438.519 
54.907} 109.815) 164.722] 219.630] 274.537] 329.444! 384.352) 439.259 
55.000) 110.000) 165.000) 220.000) 275.000] 330.000} 385.000] 440.000 
55.093) 110.135) 165.278} 220.370) 275.463] 330.556] 385.648] 440.74] 
55.185) 110.370) 165.556} 220.741! 275.926] 331.111] 386.296) 441.48] 
55.278] 110.556) 165.833] 221.111) 276.389] 331.667| 386.944] 442.292 
55.370) 110.741) 166.111) 221.481) 276.852] 332.222) 387.593] 442.963 
55.463) 110.926) 166.389) 221.852) 277.315} 332.778] 388.241] 443.704 
55.556) 111.111) 166.667) 222.222) 277.778] 333.333] 388.889] 444.444 





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XXXII.—PRISMOIDAL CORRECTIONS 
CWU-YDS JFER 1007) rr) 


3 4 5 6! 7 8 


241 








Pe et ed ed fd feet et et eet pt et 
> 
ces) 
(—) 








185 247 .809 .370 482 494 
278 370 463 .906 . 648 741 
370 494 .617 741 . 864 988 
463 .617 172 .926 | 1.080} 1.2385 
556 741 . 926 PTL 2}. 296) 41481 
648 .864 | 1.080 1.296 | 1.512 | 1.728 
741 -988 | 1.235 1.481) .1.728>| 21.975 
833 | 1.111 | 1.389 1.667 | 1.944 | 2.222 
92 1.235 | 1.543 1.852 | 2.160 | 2.469 
1.019 | 1.358 | 698 2.037 | 2.377 | 2.716 





























242 


TABLE XXXII.—PRISMOIDAL CORRECTIONS 





~3 


lee} 


10. 


11. 


— 
rae) 




















CU.YDS. PER 100 FT. 



































13.179 
13.395 
13.611 
13.827 
14.043 
14.259 
14.475 
14.691 
14.907 
15.123 


15.340 
15.556 
15.772 
15.988 
16.204 
16.420 
16.636 
16.852 
17.067 
17.284 


17.500 
17.716 
17.982 
18.148 
18.364 
18.580 
18.796 
19.012 
19.228 
19.444 


19.660 
19.877 
20.093 
20.309 
20.525 
20.741 
20.957 
21.173 
21.389 
21.605 


21.821 
22.037 
22.203 
22.469 
22.685 
22.901 


25.926 





























TABLE XXXII.—PRISMOIDAL CORRECTIONS 243 
CU.YDS. PER 100 FT. 





is 
me 





~ 


ae) 








3.785] 7.469} 11.204 | 14.938 | 18.673 | 22.407 | 26.142 | 29.877 | 33.611 
3.766] 7.531! 11.296 | 15.062 | 18.827 | 22.593 | 26.359 | 30.123 | 33.889 
3.796) .7.593' 11.389 | 15.185 | 18.981 | 22.778 | 26.574 | 30.370 | 34.167 
3.827] 7.654) 11.481 | 15.309 | 19.136 | 22.963 | 26.790 | 30.617 | 34.444 
3.858] 7.716} 11.574 | 15.482 | 19.290 | 23.148 | 27.006 | 30.864 | 34.722 
3.889] 7.778) 11.667 | 15.556 | 19.444 | 23.333 | 27.222 | 31.111 | 35.000 
3.920) 7.840) 11.759 | 15.679 | 19.599 | 23.519 | 27.438 | 31.359 | 35.278 
3.951] 7.901) 11.852 | 15.802 | 19.753 | 23.704 | 27.654 | 31.605 | 35.556 
3.981) 7.963) 11.944 | 15.926 | 19.907 | 23.889 | 27.870 | 31.852 | 35.883 
4.012} 8.025} 12.037 | 16:049 | 20.062 | 24.074 | 28.086 | 32.099 | 36.111 


4.043} 8.086) 12.130 | 16.173 | 20.216 | 24.259 | 28.302 | 32.346 | 36.389 
4.074} 8.148] 12.222 | 16.296 | 20.370 | 24.444 | 28.519 | 32.593 | 36.667 
4.105} 8.210) 12.315 | 16.420 | 20.525 | 24.630 | 28.785 | 32.840 | 36.944 


~ | 


_ 
ie) 


, 





— 
= 


4.136; 8.272] 12.407 | 16.548 | 20.679 | 24.815 | 28.951 | 33.086 | 37.222 





4.321] 8.642] 12.963 | 17.284 | 21.605 | 25.926 | 30.247 | 34.568 38. 889 


4.352] 8.704] 13.056 | 17.407 | 21.759 | 26.111 | 30.463 | 34.815 | 39.167 
4.383] 8.766} 13.148 | 17.531 | 21.914 | 26.296 | 30.679 | 35.062 | 39.444 


4 444| 8.899} 13.333 | 17.778 | 22.222 | 26.667 | 31.111 | 35.556 | 40.000 
4.475| 8.951| 13.426 | 17.901 | 22.377 | 26.852 | 31.827 | 35.802 | 40.278 


4.568) 9.136] 13.704 | 18.272 | 22.840 | 27.407 | 31.975 | 36.543 | 41.111 


4.599] 9.198] 13.796 | 18.395 | 22.994 | 27.593 | 32.191 | 36.790 | 41.389 
4.630) 9.259| 13.889 | 18.519 | 28.148 | 27.778 | 32.407 | 37.087 | 41.667 


4.660] 9.321} 13.981 | 18.642 | 23.302 | 27.963 | 32.623 | 37.284 | 41.944 


_ 
On 





\ 


I 
for) 


_ 
= 
SOCOM EN RWI SOOIRONARWUOH SODOUIBAMPWWE CODUIRONMRWUH COBDYRTIRWWH CODYIRSORwweD 


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co 





4.691] 9.383) 14.074 | 18.766 | 28.457 | 28.148 | 32.840 37.531 | 42.222 











4 938| 9.8771 14.815 | 19.753 | 24.691 | 29.630 | 34.568 | 39.506 | 44.444 
4.969/ 9.938] 14.907 | 19.877 | 24.846 | 29.815 | 34.784 | 39.753 | 44.722 








5 247110.494| 15.741 | 20.988.) 26.235 | 31.481 | 36.728 | 41.975 | 47.222 
5.278110.556| 15.833 | 21.111 | 26.389 | 31.667 | 36.944 | 42.222 | 47.500 























5 5BG/11.111| 16.667 | 22.222 | 27.778 | 33.333 | 38.889 | 44. 444 50.000 





244. TABLE 


XXXII.—PRISMOIDAL CORRECTIONS 
CU.YDS. PER 100 FT. 








cs) 


1 


ip 
= 














wis) 
© | 


5.586)11.173 
5.617/11.285 
5.648/11.296 
5.679/11.358 
5.710)11.420 
5.741/11.481 
5.772|11.548 
5. 802/11. 605 
5. 833)11.667 
5. 864/11. 728 


5. 895|11.'790 
5. 926/11. 852 
5.957/11.914 
5.988)11.975 
6.019)12.037 
6.049/12.099 
6.080)12.160 
6.111/12.222 
6.142)12.284 
6.173)12.346 


6.204/12.407 
6. 235/12. 469 
6. 265)12.531 
6.296|12.593 
6.327)12.654 
6.359) 12.716 
6.389) 12.778 
6. 420/12. 840 
6.451|12.901 

.481|12. 968 


6.512)13.025 
6.543) 13.086 
6.574)13.148 


— 
© 


no 
SOOUIRBNPWOH SOWOUIRTMPWDWH DOKOUIMMARWNWH SOHDOUARTARWHH CODNRMAWwH CBODOIOOARWHHY 

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<5 


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(=) 





6.790|13.580 
6. 821/13. 641 


na 
no 


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wo 








BS 








16.759 | 22.346 | 27.932 | 33.519 | 39.105 | 44.691 
16.852 | 22.469 | 28.086 | 33.704 | 39.321 | 44.938 
16.944 | 22.593 | 28.241 | 33.889 | 39.537 | 45.185 
17.037 | 22.716 | 28.395 | 34.074 | 39.753 | 45.432 
17.130 | 22.840 | 28.549 | 34.259 | 39.969 | 45.679 
17.222 | 22.963 | 28.704 | 34.444 | 40.185 | 45.926 
17.315 | 23.086 | 28.858 | 34.630 | 40.401 | 46.173 
17.407 | 23.210 | 29.012 | 34.815 | 40.617 | 46.420 
17.500 | 23.333 | 29.167 | 35.000 | 40.883 | 46.667 
17.593 | 28.457 | 29.321 | 35.185 | 41.049 | 46.914 


17.685 | 23.580 | 29.475 | 35.370 | 41.265 | 47.160 
17.778 | 23.704 | 29.630 | 35.556 | 41.481 | 47.407 
17.870 | 23.827 | 29.784 | 35.741 | 41.698 | 47.654 
17.963 | 23.951 | 29.938 | 35.926 | 41.914 | 47.901 
18.056 | 24.074 | 30.093 | 36.111 | 42.130 | 48.148 
18.148 | 24.198 | 30.247 | 36.296 | 42.346 | 48.395 
18.241 | 24.821 | 30.401 | 36.481 | 42.562 | 48.642 
18.333 | 24.444 | 30.556 | 36.667 | 42.778 | 48.889 
18.426 | 24.568 | 30.710 | 36.852 | 42.994 | 49.136 
18.519 | 24.691 | 30.864 | 37.037 , 48.210 | 49.383 


18.611 | 24.815 | 31.019 37.222 | 43.426 | 49.630 
18.704 | 24.938 | 31.173 | 37.407 | 48.642 | 49.877 
18.796 | 25.062 | 31.327 | 37.593 | 43.858 | 50.123 























19.444 | 25.926 32 253 38.889 | 45.370 | 51.852 
19.537 | 26.049 | 32.562 | 39.074 | 45.586 | 52.099 


19.907 | 26.543 | 33.179 | 39.815 | 46.451 | 53.086 


20.093 | 26.790 | 33.488 | 40.185 | 46.883 | 53.580 
20.185 | 26.914 | 33.642 | 40.370 |-47.099 | 53.827 


20.370 | 27.160 | 33.951 | 40.741 | 47.531 | 54.321 
20.463 | 27.284 | 34.105 | 40.926 | 47.747 | 54.568 


























& 


~ 


wD 
) 


s 


TABLE XXXII.—PRISMOIDAL CORRECTIONS 245 
CU.YDS. PER 100 FT. 





1 2 3 4 5 6 7 8 ss 

















7.438/14.877| 22.315 | 29.753 | 37.191 | 44.630 | 52.068 | 59.506 | 66.944 
7.469|14.938] 22.407 | 29.877 | 387.346 | 44.815 | 52.284 | 59.753 | 67.222 
7.500/15.000} 22.500 | 30.000 | 37.500 | 45.000 | 52.500 | 60.000 | 67.500 
7.531/15.062} 22.593 | 30.123 | 37.654 | 45.185 | 52.716 | 60.247 | 67.778 
7.562|15.123} 22.685 | 30.247 | 37.809 | 45.370 | 52.982 | 60.494 | 68.056 
7.593/15.185| 22.778 | 30.3870 | 37.963 | 45.556 | 53.148 | 60.741 | 68.333 
7.623)15.247| 22.870 | 30.494 | 38.117 | 45.741 | 53.364 | 60.988 | 68.611 
7.654/15.3809] 22.963 | 30.617 | 38.272 | 45.926 | 58.580 | 61.2385 | 68.889 
7.685|15.370) 23.056 | 30.741 | 38.426 | 46.111 | 53.796 | 61.481 | 69.167 
7.716|15.482) 23.148 | 30.864 | 38.580 | 46.296 | 54.012 | 61.728 | 69.444 


7.747\15.494| 23.241 | 30.988 | 38.785 | 46.481 | 54.228 | 61.975 | 69.722 
7.778)15.556|} 23.333 | 31.111 | 38.889 | 46.667 | 54.444 | 62.222 | 70.000 
7.809|15.617| 23.426 | 31.235 | 39.043 | 46.852 | 54.660 | 62.469 | 70.278 
7.840|15.679| 23.519 | 31.358 | 39.198 | 47.037 | 54.877 | 62.716 | 70.556 
7.870/15.741] 23.611 | 31.481 | 39.352 | 47.222 | 55.098 | 62.963 | 70.833 
7.901|15.802) 23.704 | 31.605 | 39.506 | 47.407 | 55.309 | 63.210 | 71.111 
7.932/15.864| 23.796 | 31.728 | 39.660 | 47.593 | 55.525 | 63.457 | 71.389 
7.963/15.926|} 23.889 | 31.852 | 39.815 | 47.778 | 55.741 | 63.704 | 71.667 
7.994/15.988| 23.981 | 31.975 | 39.969 | 47.963 | 55.957 | 63.951 | 71.944 
8.025)16.049| 24.074 | 32.099 | 40.123 | 48.148 | 56.178 | 64.198 | 72.222 


8.056/16.111} 24.167 | 32.222 | 40.278 | 48.333 | 56.389 | 64.444 | 72.500 
8.086/16.173| 24.259 | 32.346 | 40.432 | 48.519 | 56.605 | 64.691 | 72.778 
8.117|16.235|} 24.852 | 32.469 | 40.586 | 48.704 | 56.821 | 64.938 | 73.056 
8.148/16.296| 24.444 | 32.593 | 40.741 | 48.889 | 57.037 | 65.185 | 73.333 
8.179) 16.358] 24.537 | 32.716 | 40.895 | 49.074 | 57.253 | 65.432 | 73.611 
8.210)16.420| 24.630 | 32.840 | 41.049 | 49.259 | 57.469 | 65.679 | 73.889 
8.241/16.481| 24.722 | 32.963 | 41.204 | 49.444 | 57.685 | 65.926 | 74.167 
8.272|16.543| 24.815 | 33.086 | 41.359 | 49.630 | 57.901 | 66.178 | 74.444 
8.302/16.605} 24.907 | 33.210 | 41.512 | 49.815 | 58.117 | 66.420 | 74.722 
8.333)16.667| 25.000 | 83.333 | 41.667 | 50.000 | 58.333 | 66.€67 | 75.000 


8.364) 16.728} 25.093 | 33.457 | 41.821 | 50.185 | 58.549 | 66.914 | 75.278 
8.395|16.'790} 25.185 | 33.580 | 41.975 | 50.370 | 58.766 | 67.160 | 75.556 
8.426)16.852| 25.278 | 33.704 | 42.130 | 50.556 | 58.981 | 67.407 | 75.833 
8.457/16.914| 25.370 | 33.827 | 42.284 | 50.741 | 59.198 | 67.654 | 76.111 
8.488/16.975| 25.463 | 33.951 | 42.488 | 50.926 | 59.414 | 67.901 | 76.389 
8.519|17.037| 25.556 | 34.074 | 42.593 | 51.111 | 59.630 | 68.148 | 76.667 
8.549'17.099} 25.648 | 34.198 | 42.747 | 51.296 | 59.846 | 68.395 | 76.944 
8.580)17.160| 25.741 | 34.321 | 42.901 | 51.481 | 60.062 | 68.642 | 77.222 
8.611/17.222| 25.833 | 34.444 | 43.056 | 51.667 | 60.278 | 68.889 | 77.500 
8. 642|17.284| 25.926 | 34.568 | 43.210 | 51.852 | 60.494 | 69.186 | 77.778 


8.673/17.346} 26.019 | 34.691 | 43.364 | 52.037 | 60.710 | 69.383 | 78.056 
8.704)17.407] 26.111 | 34.815 | 43.519 | 52.222 | 60.926 | 69.630 | 78.383 
8.735/17.469] 26.204 | 34.938 | 43.673 | 52.407 | 61.142 | 69.877 | 78.611 
8.766|17.531| 26.296 | 35.062 | 438.827 | 52.593 | 61.359 | 70.128 | 78.889 
8.796|17.593) 26.389 | 35.185 | 43.981 | 52.778 | 61.574 | 70.370 | 79.167 
8.827|17.654| 26.481 | 35.309 | 44.136 | 52.963 | 61.790 | 70.617 | 79.444 
8.858|17.716] 26.574 | 35.432 | 44.290 | 53.148 | 62.006 | 70.864 | 79.722 
8.889|17.778] 26.667 | 35.556 | 44.444 | 53.333 | 62.222 | 71.111 | 80.000 
8.920/17.840| 26.759 | 35.679 | 44.599 | 538.519 | 62.438 | 71.359 | 80.278 
8.951/17.901] 26.852 | 35.802 | 44.753 | 53.704 | 62.654 | 71.605 | 80.556 


8.981/17.963) 26.944 | 35.926 | 44.907 | 53.889 | 62.870 | 71.852 | 80.833 
9.012/18.025| 27.037 | 36.049 | 45.062 | 54.074 | 63.086 | 72.099 | 81.111 
9 .043/18.086| 27.130 | 36.173 | 45.216 | 54.259 | 63.302 | 72.346 | 81.389 
9.074|18.148} 27.222 | 36.296 | 45.370 | 54.444 | 63.519 | 72.593 | 81.667 
9.105|18.210} 27.315 | 36.420 | 45.525 | 54.630 | 63.7385 | 72.840 | 81.944 
9.136|18.272| 27.407 | 36.543 | 45.679 | 54.815 | 68.951 | 73.086 | 82.222 
9.167|18.333] 27.500 | 36.667 | 45.833 | 55.000 | 64.167 | 73.333 | 82.200 
9.198/18.395| 27.593 | 36.790 | 45.988 | 55.185 | 64.383 | 73.580 | 82.778 
9 .228/18.457| 27.685 | 36.914 | 46.142 | 55.370 | 64.599 | 73.827 | 83.056 


= | 


2 n ‘ vn raw) 
Le) co —I for) R 


SOCOM EMR WDE SOOIRAONRWMWH DOWDUIRORWNOH DODARNUPWHOH DODAIRAMNPWHH DODNIRORWmoe 


ow 
o 






































9.259118.519| 27.778 | 37.037 | 46.296 | 55.556 | 64.815 | 74.074 | 83.333 


246 TABLE XXXIII.—MINIMUM LENGTH OF SPIRAL 
CURVE 


Minimum Length of Spiral in Feet 







































































































































































0 106 200 300 400 500 
\ 
i Th Sb est 
90 rer Oss St 
& % 
4418 LI hat 
8 80 an 7 os 
+Oy; I} + +—l—t 
: : ed & 
8 oT ST ear Rear ee 
ord + || | C | ree Oa 
< 3 @. KS 
= 1 rg, Z Ss cs 
& 60H a coy ou 
“T1 
3 xo 
g 3 PS " 
mM 50 r 0 ral 
® 2 
a a 6 
Bors 5 
£ 5 8 
“it BerT |S 
Aa 30 8 Rea] A 
4 2 A PTT |S 
— 20 2 i 3 
| oy 5 
‘ Gai A) Ps 
10 a | Ey 
R 
60 100 200 


Limiting Curves. For all curves which are liable to limit 
the speed of trains, the length of spiral should equal that indi- 
cated on the line marked “ Superelevation =8 inches.”’ Longer 
spirals may be used provided the increased length does not 
adversely affect the degree of curve or seriously affect the cost 
of construction. 

Minor Curves. For minor curves the length of spiral 
should never be less than that indicated by the diagram; an 
increase of about 50 per cent over the indicated length may be 
desirable where cost is not seriously affected. 

Spirals need not be used when superelevation required for 
highest permissible speed is less than two inches. 


TABLE XXXIV.—ELEMENTS OF THE SPIRAL OF 247 
CHORD-LENGTH, 100 










































































Degree Spiral (Inclination) Latitude of Sum of the 
n of Curve Angle. jof Chord to} Each Chord. Latitudes. 
Ds 3 Axis of Y. | 100 Xcos Incl. y 
0 0° 00’ 0° 00’ 0° 00’ 
] 10 10 05 99 . 99989423 99. 99989423 
2 20 30 20 99 . 99830769 199 . 99820192 
3 30 1 45 99 . 99143275 299 . 98963467 
4 40 1 40 devel) 99. 97292412 399 . 96255879 
5 50 2 30 2 05 99 . 93390007 499 . 89645886 
6 Le 3 30 3 99 . 8629535 599 . 7594123 
7 1 10 4 40 45.05 99 . 7461539 699. 5055662 
8 deo 6 ie tose 20) 99 . 5670790 - 799. 0726452 
9 1 30 7 30 6 45 99. 3068457 898. 3794909 
10 1 40 ee 8 20 98. 944164 997 . 3236549 
ll 150 ll 10 - 05 98. 455415 1095. 779070 
12 2 13 12 97. 814760 1193 . 593830 
13 2) 10 15 10 ~14 05 96 . 994284 1290. 588114 
14 2 20 17. 30 16m 20 95. 964184 1386. 552298 
15 2 30 20 18 45 94.693014 1481 . 245312 
16 2 40 22 40 eLe20 93. 147975 1574. 393287 
17 2, 50 20 30 24 05 91. 295292 1665. 688579 
18 3a) 28 30 at 89. 100650 1754. 789229 
19 3 10 381 40 30 05 . 86.529730 1841. 318959 
20 3 20 385 33 20 83. 548780 1924. 867739 
- Departure of Sum of the : . 
n Each Chord. Departures. Logarithm. Logarithm. 
100 Xsin Incl. ab log y log x 
1 . 1454441 . 145444] 1. 9999995 9. 1626960 
2 . 5817731 eielelia 2.3010261 9. 8616641 
3 1. 3089593 2.0361765 2. 4771063 0. 3088154 
4d 2. 38268960 4. 3630725 2.6020194 0.6397924 
5 3. 6353009 7. 9983734 2. 6988800 0. 9030017 
6 5 : 233596 13. 231969 PPO AEAL 1.1216244 
fi 7.120730 20. 352699 2. 8447911 1.3086220 
tae 9.294991 29. 647690 2. 9025862 1.4719909 
9 11. 753874 41.40143 2. 9534598 1.6170153 
10 14. 49319 55 . 89462 2. 9988361 1. 7473701 
ll 17. 50803 73. 40265 3. 0397231 1.8657117 
+12 20. 79117 94.19382 3. 0768567 1.9740224 
13 24. 33329 118.52711 3.1107877 2.07388177 
14 28.12251 146. 64962 3. 1419362 2. 1662811 
15 32. 14395 178. 79357 3. 1706269 2.20e8019 
16 36. 37932 215.17289 3.1971131 2.8327875 
17 40. 80649 255. 97938 . 8.2215938 2. 4082049 
18 45 . 39905 301.37848 3. 2442250 2.4791121 
19 50.12591 351. 50434 3.2651291 2.5459307 
20 54. 95090 406. 45524 3. 2844009 2. 6090128 
ae Deflection ees Deflection 
n Log y Angle. Rs n Log y Angle. Rs 
log tan 7 1 log tan 7 D 
1 |7.1626964 |0° 05’ 00.’" 00 /34377.5 11 |8.8259886 | 3° 49’ 56.’”” 39 | 3125.36 
2 |7.5606380 |0°-12’ 30.” 00 |17188.8 12 |8.8971657 | 4° 80’ 43.” 95 | 2864.93 
3 |7.8317091 |0° 23’ 20.”” 00 |11459.2 13 |8.9680300 | 5° 14’ 50.” 28| 2644.98 
4 |8.0377730 |0° 37’ 29.”” 99 | 8594.42}! 14 |9.0243449 | 6° 02’ 14.’” 93| 2455.70 
5 |8.2041217 |0° 54’ 59." 97 | 6875.55}| 15 .|9.0817250 | 6° 52’ 57.’ 31 | 2292.01 
6 |8.3436478 1° 15/49." 90 | 5729.65]; 16 |9.13856744 | 7° 46’ 56.” 71 | 2148.79 
7 |8.4638309 |1° 39’ 59.’ 75 | 4911.15!} 17 |9.1866111 | 8° 44’ 12.” 26| 2022.41 
8 |8 5694047 |2° 07’ 29.”" 45 | 4297.28) 18 |9.2348871 | 9° 44’ 42.’” 92} 1910.08 
9 |8.6635555 |2° 38’ 18.’” 90 | 3819.83}; 19 |9.2808016 |10° 48’ 27.’ 44] 1809.57 
10 |8.7485340 |3° 12’ 27.”’ 95 | 3487.87/| 20 |9.38246119 111° 55’ 24.”” 34} 1719.12 




















nr et 


248 


Rule for Finding a Deflection 


TABLE XXXV.—DEFLECTION ANGLES, FOR 


Read under the heading corresponding to the point at which the instru- 


ment stands, and on the line of the number of the point observed. . 


WOOND OPWWHO | 3 


10 








oO 
° 
° 
oO 





mow c = 3 tae pie WWWrFr- 
COO NOOO WMDHe 
<© 00 AOS Ee WWM 
cco AOL WWwOH eH 


so} 


Inst. at 3 


40’’ 


30 


eco NOOPCS WORF 


10 





| 


_ 
SCMOND OP WWE SO 


Inst. at 8 





3° 52’ 3 





eS RwmMWwH 


_ 
SCoOeo NAOOewS Dee 





ok 
Wr OOO SRMOPCO WORE 
— 
— 


Wr OOF MDOPwwO 


— 
Cw 
— 








FeO WWW 


NWNoOeCom MAP WwWwrR 


pad pe 








Oconto OP WMWrRO 


10 


LOCATING SPIRAL CURVES IN THE FIELD 


Rule for Finding a Deflection 


249 


Read under the heading corresponding to the point at which the instru- 
ment stands, and on the line of the number of the point observed. 





i) 
SOOND OPWWrHO | 3 


| 


WOOD OP WoOrHoO 


10 
































Inst. at 10/Inst. at 11/Inst. at 12|Inst. at 13) Inst. at 14 | Inst. at 15 
a a a a t 4 ue 
Boe noe! | T° 510% 04/7) 8° 297 16/7159° 5b” 10745 11S 277 -45/7| 13°°077 03%) 0 
5 36 424-6 47 33 | 8 05 05 | 9 29 18 | 11 00 13 | 12 37 49 1 
Dts BOL oO elas42 107 83 (840 1099900) O6iaiel Oe 29) 2041) 1205 e16 2 
Bb WS KOE | oy) GaP a 1 Oy SY etsy Pde ahs Feb5) OS BL 4298 23 3 
Aeete el 180) ZOMOL \FG.%32 Senin Oly 44 OR ver soul), LOs50n110 4 
Smee BOO (4 44010 151.55" Ole ve 12) 32 8 36:45 | 10. 07 37 Ii 
se0e mol it 05.200 | 5 14: dds 6 “307 02 (Be BB OF 218 45 6 
ceele40 ES. aeano0) 174: 630). .00) npr 44) TE] 7 05 02 Cy pry) Bl 7 
1537 730 he 386 40) 1238) G42 30) 45551 00 (ye, We 7 40 02 8 
50 1 4% 680"1°2 (51540) (p45 02" 30 5 20 00 6 44 ll 9 
00 55 L5% ya0h ies 06) 40 4 22 30 5 45 01 | 10 
55 00 17700" 00) bias 07% 30 3 21 40 4 42: 30 | Il 
1652 830- iF 1 -00:.00 00 1 05 00 PAB) 3 36 40 | 12 
2 Oo meeLO beaks B0niel . 05. 06 00 1 10 00 Cac OUL els 
SreOeso0 jbo “OSiecOn 2 1 12 302 ee o10) 00 00 Ly Td 00K 14 
Hee0H e004 iano) Oereacale lmoe ee, 30 1 15 00 00 15 
perso: 529.0598 164. 37 130) Ind 988° 20 1) oa) BU Ly 20; 0054 16 
eed WoT tl 6 40248.) 5 154 59 Wa 57 30 3: 538 20 Cae OOM LT 
8 47 2418 04 57 | 7 15 48 |6 19 59 oe lieol 4 08 20 | 18 
10 08 10 | 9 27 24 | 8 39 56 |7 45 48 6 44 59 By ef SBOE WIRY 
ine soel 4 HOM ba 09M LOs 07 = 2a Om 14) 56 8 15 48 7 0959 -|-.20 

Inst. at 16 | Inst. at 17 | Inst. at 18 | Inst. at 19 | Inst. at 20 
t om, a t 4 Ww 
Ace ame OD Gon 4) 6484) 18nd 5 ealy 2 b20S W383 ls 03% 04. 36710 0 
14 22 09 16. 13. Il 18 10 59 20) 7 15s 32 Cem eom De 1 
13 47 54 Lio! 715 Kies Boel 19 36 ll 21 45 48 2 
T3/5,10...20 14. 57 -59 16m 5eueco JIS GB) 34 Paleo OM a5} 3 
12 29 26 EE Shy ee! 16 08 05 18 O07 31 20m tome 4 
tees 212 13 29 29 Tom cme Lime lSeel? 19 22 40 5 
10 57 39 12 40 14 ILD PA) BP l6mcouneo 18 28 19 6 
10 06 46 lea 4) TSigcde aly. 15 29 36 17 30 39 7 
Ore Zp 54 Tepe 47 IP Bip eee 14 30 20 16 29 40 8 
8 15. 03 Sr) Bi ll 36 49 TS a4 1519-25) 23 9 
i 14 ik 8 50 03 10 32 36 12° 21 -50 14 17 46 10 
6 10 Ol tae 4425 12 9.5 25 03 Tit 236 13 06 51 ll 
5 02; 30 Oa oome OL 8 14 12 10 00 04 VL By Bie 12 
a5 40 Ber de; 30 7 00 Ol ease le 10 35 04 13 
2 BY RR, 4 06 40 5 42430 eon OF Qe t4 12 14 
ti 20° 00 er Al 30 4~ 21° 40 6 02 30 ae DOM OL 15 
00 1 25 00 aot. a0 4 36 40 6 22 30 16 
1 25: 00 00 1 30 00 3 O07 30 AeA () 17 
Py BN) 1) 30) 00 00 Teas a. LT 30 18 
4°23 20 3°02 30 1-35. 00 00 1 40 19 
Dae Diol 4 38 20 3) 22, 30 140 00 20 








250 TABLE XXXVI.—DEGREES OF CURVES 


DEGREE OF CURVE AND VALUES. OF THE COORDINATES & AND Y, FOR EACH 
CHOoRD-POINT OF THE SPIRAL FOR VARIOUS LENGTHS OF THE CHORD 


. CHORD-LENGTH = 10 















































n ne Ds Yy x Log x 
1 10 Ted. 00% 10.000 0.0145 8.162696 
2 20 3 20 02 20.000 0727 8.861664 
3 30 5 00 06 29.999 . 2036 9.308815 
4 40 6 40 138 39.996 . 4363 9.639792 
5 50 8220.26 49.990 7998 9.903002 
6 60 10° 00 45 59.976 1.323 0.121624 
ib 70 IT 47 12 69.951 2.085 0.308622 
8 80 13 21 48 79.907 2.965 0.471991 
9 90 15.02) 34 89.838 4.140 0.617015 
10 100 16 43 31 99.732 5.589 0.747370 
11 110 WS 24042 109.578 7.340 0.865712 
12 120 20 06 07 119.359 9.419 0.974022 
13 130 21 47 48 129.059 11.853 1.073818 
14 140 23 29 46 138.655 14.655 1.166281 
15 150 25. Lese02 148.125 17.879 1.252852 
16 160 26 54 39 157.439 BIST 1.332788 
iN 170 28 387 38 166.569 25.598 1.408205 
18 180 30 21 Ol 175.479 30.138 1.479112 
19 190 382 04 48 184.1382 35.150 1.545931 
20 200 33 49 02 192.487 40.646 1.609018 
35 33 46 
c. CHORD-LENGTH = 11 
n ne Ds Yy x Log x 
1 ll Se 2557 11.000 0.0169 8.204089 
2 22 3 01 50 22.000 . 0800 8.908057 
3 33 4 382 48 82.999 .2240 9.350208 
4 44 6 03 48 43.996 4799 9.681185 
5 55 7 34 52 54.989 8798 9.944394 
6 66 9 06 Ol 65.974 1.456 0.163017 
Ve 77 10 37 16 76.946 2.2389 0.350015 
8 88 12 08 37 87.898 8.261 0.513384 
9 99 13 40 06 98 . 822 4.554 0.658408 
10 110 15 It 44 109.706 6.148 0.788763 
11 121 16 438 31 120.536 8.074 0.907104 
12 132 18 15 29 181.295 10.361 1.015415 
13 143 19 47 39 141.965 13.038 1.115210 
14 154 Ar Alo 152.521 16.1381 1.207674 
15 165 22 52 38 162.937 19.667 1.293745 
16 176 24 25 29 173.188 23.669 1.374180 
17 187 25 58 36 183.226 28.158 1.449598 
18 198 el moe MOL 193.027 30. 152 1.520505 
19 209 29 05 45 202.545 88.665 1.587823 
20 220 os 3 - 211.785 44.710 1.650405 














OAD OP whore 





ro) 
fon loKo ol» for) OP WO 





nc 











AND COORDINATES % AND Y 


Cc. 


CHORD-LENGTH = 12 





y 


12.000 
24.000 
35.999 
47.996 
59.988 


71.971 
83.941 
95 . 889 
107. 806 
119.679 


131. 493 
143. 231 
154. 871 
166.386 
177.749 


188.927 
199. 883 
210.575 
220.958 
230.984 





CHORD-LENGTH = 13 








y 


13.000 
26.000 
38.999 
51.995 
64.987 


77.969 
90.936 
103.879 
116.789 
129. 652 


142.451 
155. 167 
167.776 
180.252 
192.562 


204.671 
216:540 
228.123 
239.371 
250.233 








201 





Log zx 


8.241877 
8.940845 
9.387997 
9.718974 
9.982183 


0. 200806 
0.387803 
0.551172 
0.696196 
0.826551 


0.944893 
1.053204 
1.152999 
1.245462 
1.331533 


1.411969 
1. 487386 
1.558293 
1.625113 
1.688194 


Log x 


8.276639 
8.975607 
9. 422759 
9.753736 
0.016945 


0. 235568 
0. 422565 
0.585934 
0.730959 
0.861313 


0.979655 
1.087966 
1.187761 
1.280224 
1.366295 


1.446731 
1.522148 
1.593055 
1.659874 
1.722956 


252 





TABLE XXXVI.—DEGREES OF CURVES 


Cc. 


CHORD-LENGTH = 14 























n ne D, y 

iy 14 oe Lieeby 14.000 
2 28 eames, Oc 28.000 
3 42 3 34 19 41.999 
4 56 4 45 48 55.995 
5 70 Hieo7 18 69. 986 
6 84 7208 51 83.966 
7 98 8 20 26 97.9381 
8 112 9 32 04 111.870 
9 126 10 48 47 125.773 
10 140 ll 55 33 139.625 
1 154 13°07) 24 153. 409 
12 168 14 19 20 ‘167.103 
13 182 vessil 27 180.682 
14 196 16 43 29 194.117 
15 210 17 55 44 207 .3874 
16 224 19 06 05 220.415 
17 238 20 20 34 233.196 
18 252 ray ay atl 245.670 
19 266 22 45 56 257.785 
20 280 23 58 51 269.481 

Pay JL ye 
ec. CHORD-LENGTH = 15 

n nc D, Yy 

31 15 1° 06’ 40” 15.000 
2 30 2 13 20 30.000 
3 45 3 20 02 ' 44.998 
4 60 4 26 44 59.994 
5 75 5 33 28 74.984 
6 90 6 40 13 89.964 
ff 105 po YE AO 104.926 
8 120 8 53 51 119.861 
9 135 10 00 45 134.757 
10 150 Lip e41 149.599 
ll 165 12 14 41 164.367 
12 180 ips ale Hy 179.039 
13 195 14 28 56 193.588 
14 210 15 36 09 207 .983 
15 b220 16 43 28 222.187 
16 240 17 50 54 236.159 
17 255 psy lafex Ae 249. 853 
18 270 20 06 02 263.218 
19 285 rake Bis} Gil 276.198 
20 300 22 21 39 288.730 

23 29 48 














Log x 


8.308824 
9.007792 
9.454943 
9.785920 
0.049130 


0.267752 
0.454750 
0.618119 
0.763143 
0. 893498 


. 011840 
. 120150 
219946 
812409 
. 898480 


478915 
. 904333 
625240 
. 692059 
1.755141 


— | — pet et et et 








Log x 


8.338787 
9.037755 
9. 484907 
9.815884 
0.079093 


0.297716 
0.484713 
0. 648082 
0.793107 
0.923461 


1.041803 
1.150114 
1.249909 
1.349372 
1.428443 


1.508879 
1.584296 
1.655208 
1.722022 
1.785104 














AND COORDINATES XY AND y 


ec. CHORD-LENGTH = 16 














253 








n ne D, y x Log x 
1 16 1° 02’ 30” 16.000 0.0233 8.366816 
2 32 2 05 00 32.000 .1164 9.065784 
3 48 3 07 31 47.998 3258 9.512935 
4 64 4 10 03 63.994 6981 9.843912 
5 80 5 12 36 79.983 1.280 0.107122 
6 9°) 6 15 ll 95.961 2.117 0.325744 
7 112 7 17 47 111.921 3.256 0.512742 
8 128 8 20 26 127.852 4.744 0.676111 
9 144 9 23 07 143.741 6.624 0.821135 
10 160 10 25 51 159.572 8.943 0.951490 
ll 176 ll 28 37 175.325 11.744 1.069832 
12 192 | 12 381 28 190.975 15.071 1.178142 
13 208 13 34 2l 206. 494 18.964 1.277938 
14 224 14 37 20 221.848 23.464 1.370401 
a 240 15 40 21 236.999 28.607 1.456472 
16 256 16 43 28 251.903 34.428 1.536907 
17 ere 17 46 40 266.510 40.957 1.612325 
18 288 18 49 57 280.766 48.221 1.683232 
19 304 19 53 20 294.611 56.241 1.750051 
20 320 20 56 49 307.979 65.032 1.813133 
22 00 23: 
e. CHORD-LENGTH = 17 
n ne De y x Log x 
1 17 0° 58’ 49” 17.000 0.0247 8.393145 
2 34 1 57 38 34.000 . 1236 9.092113 
3 51 2 56 27 50.998 38461 9.539264 
4 68 38 55 19 67.994 TALT 9.870241 
5 85 4 54 12 84.982 1.360 0.188451 
6 102 5 53 06 101.959 2.249 0.352073 
WA 119 6 52 00 118.916 - 3.460 0.539071 
8 136 7 50 57 135.842 5.040 0.702440 
9 153 8 49 55 152.725 7.038 0.847464 
10 170 9 48 56 169.545 9.502 0.977819 
11 187 10 48 00 186 . 282 12.478 1.096161 
12 204 1l 47 O7 202.911 16.013 1.204471 
13 221 12 46 15 219.400 20.150 1.304267 
14 238 13 45 27 235.714 24.930 1.396730 
15 255 14 44 44 251.812 30.395 1. 482801 
16 PAC 15 44 03 267.647 36.579 1.563236 
17 289 16 438 27 283. 167 43.516 1.638654 
18 306 17 42 56 298.314 51.234 1.709561 
19 323 18 42 29 313.024 59.756 1.776380 
20 340 19 mv Hf 327.228 69.097 1.839462 
20 











254 


TABLE XXXVI.—DEGREES 


C. 


OF CURVES 


CHORD-LENGTH = 18 












































n nc P, Yy cD Log x 
1 18 0° 55’ 33” 18.000 0.0262 8.417968 
2 36 1 51 07 36.000 . 1309 9.116937 
3 54 2 46 40 53.998 . 8665 9.564088 
4 72 3 42 16 71.993 . 7853 9. 895065 
5 90 4 87 51 89.981 1.440 0.158274 
6 108 5 83 28 107.957 2.382 0.376897 
7 126 6 29 05 125.911 3.663 0.563894 
8 144 7 24 45 143.833 5.337 0.727263 
9 162 8 20 26 161.708 7.452 0.872288 
10 180 9 16 08 179.518 10.061 1.002648 
11 198 10 11 54 197.240 13.212 1.120984 
12 216 Ll 07 “41 214.847 16.955 1.229295 
13 234 12 03 31 282.306 21.335 1.329090 
14 252 12 59 24 249.579 26.397 1.421554 
15 270 13 55 20 266. 624 32.183 1.507624 
16 288 14 51 18 283.391 38.731 1.588060 
17 306 15 47 20 299. 824 46.076 1.663477 
18 324 16 48 27 315.862 54.248 1.734385 
19 342 17 39 37 331 . 437 63.271 1.801203 
20 360 18 35 51 346.476 73.161 1. 864285 
19 32 08 
c. CHORD-LENGTH = 19 
n nc D, y x Log « 
1 19 0° 52’ 38” 19.000 0.0276 8.441450 
2 38 1 45 16 38. 000 . 1382 9.140418 
3 57 2 87 54 56.998 . 8869 9.587569 
4 76 38 30 34 75.993 -8290 9.918546 
5 95 4 23 18 94.980 1.520 0.181755 
6 114 5 15 54 113.954 2.514 0.400378 
7 133 6 08 36 182.906 3.867 0.587376 
8 152 7 Ol 19 151.824 5.633 0.750744 
9 171 7 54 03 170. 692 7.866 0.895769 
10 190 8 46 49 189.491 10.620 1.026124 
11 209 9 39 36 208.198 13.947 1.144465 
12 228 10 32 26 226.783 17.897 1.252776 
13 247 ll 25 18 245.212 22.520 1.352571 
14 266 12 18 12 263.445 27.863 1.445035 
15 285 13 11 09 281.437 33.971 1.531105 
16 304 14 04 09 299.135 40. 883 1.611541 
17 323 14 57 ll 316.481 48.636 1.686958 
18 342 15 50 16 333.410 57.262 1.757866 
19 361 16 43 25 349. 851 66.786 1.824684 
20 380 + eS 365.725 77.226 1.887766 











_ 
SCMOOND OP wre 











AND COORDINATES X AND Y 


CONAN we oO 


et 
DH OOO 


° 


c. 


Cc. 


CHORD-LENGTH = 20 


205 








119.952 
139.901 
159.815 
179.676 
199. 465 


219.156 
238.719 
258.118 
277.310 
296.249 


314.879 
333.138 
350. 958 
368.264 
384.974 





0.0291 
1454 
4072 
. 8726 

1.600 


2.646 
4.071 
5.930 
8.280 


_ 11.179 


14.681 
18.839 
23.705 
29.330 
35.759 


43.035 
51.196 
60.276 
70.301 
81.290 


CHORD-LENGTH = 21 














n ne D ; 
1 2l Oh ou ie 
2 42 1> 35. 14 
3 63 2.22. 02 
4 84 3 10 30 
5 105 3 58 08 
§ 126 4 45 47 
7 147 Gaoome!. 
8 168 6 21 08 
9 189 7 08 50 
a 210 dun BO 
ll 201 8 44 18 
12 252 9 32 03 
13 278 10 19 51 
14 294 ll 07 40 
15 315 11 50-31 
16 336 12 43 24 
17 357 13 31 20 
18 378 14,19 17 
19 399 LER ely 
20 420 15 55 19 
16 48 26 








Log x 


8.463726 
9.162694 
9.609845 
9.940822 
0.204032 


0. 422654 
0.609652 
0.773021 
0.918045 
1.048400 


1.166742 
1.275052 
1.374848 
1.467311 
1.553882 


1.633817 
1.709235 
1.780142 
1.846961 
1.910043 








y x Log x 
21.000 0.0305 8.484915 
42.000 mloet 9.183883 
62.998 .4276 9.631035 
83.992 .9162 9.962012 

104.978 1 680 0.225221 
125.949 2.779 0.443844 
146.896 4.274 0.630841 
167.805 6.226 0.794210 
188. 660 8.694 0.939235 
209.438 11.738 1.069589 
230.114 15.415 1.187931 
250.655 19.781 1.296242 
271.023 24.891 1.396037 
291.176 80.796 1.488500 
811.062 87.547 1.574571 
330. 623 45.186 1.655007 
349.795 53.756 1.730424 
368 . 506 63.289 1.801331 
386.677 73.816 1.868150 

85.356 1.931232 


404.222 








256 TABLE XXXVI.—DEGREES OF CURVES 


c. CHORD-LENGTH = 22 



































n nC D, Yy zc Log x 
1 22 An” 217” 22.000 0.0320 8.505119 
2 44 1° 30 53 44.000 . 1600 9.204087 
3 66 2melGe ae 65.998 . 4480 9.651238 
4 88 SAUL 50 87.992 .9599 9.982215 
5 110 3 47 18 109.977 1.760 0.245424 
6 132 4 32 48 131.947 2.911 0.464047 
7 154 O28, 18 153.891 4.478 0.651045 
8 176 6 03 48 175.796 6.522 0.814414 
9 198 6 49 19 197.643 9.108 0.959438 
10 220 7 34 51 219.411 12.297 1.089793 
Il 242 8 20 25 241.071 16.149 1.208134 
12 264 9 06 00 262.591 20.728 1.316445 
13 286 9 51 36 283.929 26.076 1.416240 
14 308 NDE Sie WBS 805. 042 32.263 1.508704 
15 330 1GT 2) 8} 825.874 39.335 1.594775 
16 “352 12 08 34 346. 367 47.338 1.675210 
i 374 12 54 16 366.451 56.315 1.750628 
18 396 13 40 O1 386.054 66.303 1.821585 
19 418 14 25 49 405.090 77.331 1. 888353 
15 11 40 
c. CHORD-LENGTH = 23 
1 23 0° 43’ 29’” 23.000 0.0335 8.524424 
2 46 Ty.c6: 58 46.000 1673 9.223392 
3 69 2 10 26 68.998 4683 9.670543 
4 92 2 538 56 91.991 1.004 0.001520 
D 115 3. 37. 26 114.976 1.840 0.264729 
6 138 Ame) 200 137.945 3.048 0.483352 
-7 161 5 04 26 160.886 4.681 0.670350 
8 184 5 47 58 183.787 6.819 0.833719 
9 207 6 31 30 206. 627 9.522 0.978743 
10 230 7 15 04 229.384 12.856 1.109098 
11 253 7 58 38 252.029 16.883 1.227439 
12 276 8 42 18 274.527 21.665 1.335750 
13 299 9 25 49 296. 835 27.261 1.435545 
14 322 10 09 27 318.907 33.729 1.528009 
15 345 10 53 06 340.686 41.123 1.614080 
16 368 Il 36 47 362.110 49.490 1.694515 
17 391 12 20 29 883.108 58.875 1.769933 
18 414 13 04 13 403. 602 69.317 1.840840 
13° 47 59 
c. CHORD-LENGTH = 24 
1] 24 Al’ 40” 24.000 0.0349 8.542907 
2 48 Ie 2dm20 48.000 1745 9.241875 
3 72 2 05 00 71.998 4887 9.689027 
4 96 2 46 41 95.991 1.047 0.020004 
5 120 3 28 22 119.975 1.920 0.283213 
6 144 4 10 03 143.942 3.176 0.501836 
7 168 4 51 45 167.881 4.885 0.688833 
8 192 5 33 28 191.777 Told 0.852202 
9 216 6 15 10 215.611 9.936 0.997226 
10 240 6 56 54 239.358 13.415 1.127581 
ll 264 7 38 39 262.987 17.617 1.245923 
12 288 8 20 25 286.463 22.607 1.354234 
13 312 9 02 12 309.741 28.446 1.454029 
14 336 9 44 00 332.773 35.196 1.546492 
15 360 10 25 48 305.499 42.910 1.632563 
16 384 ll 07 39 377. 854 51.641 “1.712999 
17 408 ll 49 31 399.765 61.435 1.788416 











AND COORDINATES XY AND Y 


c. 


CHORD-LENGTH = 25 








7 
































n ne D, Yy x Log x 
1 25 0° 40’ 00’ 25.000 0.0364 8.560636 
2 50 1 20 00 50.000 1818 9. 259604 
3 "5 2 00 00 14.997 “5090 9.706755 
4 100 2 40 Ol 99.991 1.091 0. 087732 
5 125 3 20 02 124.974 2.000 0.300942 
6 150 4 00 03 149.940 3.308 0.519564 
7 175 4 40 04 174.876 5.088 0.706562 
8 200 5 20 06 199.768 7.412 0.869931 
9 295 6 00 09 224 595 10.350 1.014955 
10 250 6 40 13 949 331 13.974 1.145310 
11 275 7 20 17 273.945 18.351 1.263652 
12 300 8 00 22 298. 398 23 548 1.371962 
13 395 8 40 28 322. 647 29.632 1.471758 
14 350 9 20 35 346. 638 36. 662 1.564221 
15 375 10 00 43 370.311 44. 698 1.650292 
16 400 10 40 52 393.598 53.793 1.730727 
17 425 ll 21 03 416.421 62.995 1.806145 
12 01 16 
ec. CHORD-LENGTH = 26 
1 26 0° 38’ 28” 26.000 0.0378 8.577669 
2 52 1 16 56 52.000 1891 9. 276637 
3 78. 1 55 24 7.997 5294 9.723789 
4 104 2 33 52 103.990 1.134 0.054766 
5 130 3 12 20 129.973 2.080 0.317975 
6 156 3 50 48 155.937 3.440 0.536598 
7 182 4 29 18 181.871 5.292 0.723595 
8 208 5 07 48 207 .'759 7.708 0.886964 
9 934 5 46 18 933 579 10.764 1.031989 
10 260 6 24 48 259.304 14.533 1.162343 
ll 286 7 03 20 284.908 19.085 1.280685 
12 312 ” 41 52 310.334 24.490 1.388996 
13 338 8 20 25 335553 30.817 1.488791 
14 364 8 58 59 360.504 38.129 1.581254 
15 390 9 37 33 385.124 46.486 1.667325 
16 416 10 16 09 409.341 55. 945 1.747761 
10 54 47 
c. CHORD-LENGTH = 27 
1 a7 0° 37’ 02” | 27.000 | 0.0393 8.594060 
2 54 1 14 04 54.000 1963 9. 293028 
3 81 1 51 07 80.997 5498 9.740179 
4 108 2 28 10 107.990 1.178 0.071156 
5 135 3 05 12 134.972 2.160 0.334365 
6 162 3 42 15 161.935 3.573 0.552988 
7 189 4 19 19 188.866 5.495 0.739986 
8 216 4 56 23 215.750 8.005 0.903355 
9 243 5 33 28 942562 11.178 1.048379 
10 270 6 10 32 269.277 15.092 1.178734 
ll 297 6 47 38 295. 860 19.819 1.297075 
12 324 7 24 44 329 270 95 432 1.405386 
13 351 8 ‘01 51 348. 459 32.002 1.505181 
14 378 8 38 59 374.369 39.595 1.597645 
15 405 9 16 07 399. 936 | 48.274 1.683716 





258 














TABLE 





Cc. 


XXXVI.—DEGREES 











OF CURVES 


CHORD-LENGTH = 28 




















n nc Ds y x Log x 
1 28 0° 35’ 42” 28.000 0.0407 8.609854 
2 56 ihe ST ts} 55.999 . 20386 9.308822 
3 84 1 47 08 83.997 5701 9.755973 
4 112 2. ee 02 111.990 1.222 0.086950 
5 140 2 58 36 139.971 2.240 0.350160 
6 168 3 34 19 167.933 3.705 0.568782 
ib 196 4 10 03 195.862 5.699 0.755780 
8 224 4 45 48 223.740 8.301 0.919149 
9 252 Daechese 251.546 11.592 1.064173 
10 280 ye ye AM 279.251 15.650 1.194528 
11 308 6 33 03 306.818 20.553 1.312870 
12 336 7 08 50 334.206 26.374 1.421180 
13 364 7 44 36 361.365 33.188 1.520976 
14 392 8 20 24 388. 235 41.062 1.613439 
15 410 8 56 13 414.748 50.062 1.699510 

9 32 04 

c. CHORD-LENGTH = 29 

1 29 0° 34’ 29” 29.000 0.0422 8.625094 
2 58 1 08 58 57.999 .2109 9.324062 
3 87 1 48 27 86.997 5905 9.771213 
4 116 Cael DO 115.989 1.265 0.102190 
5 145 2 52 26 144.970 2.320 0.365400 
6 174 3 26 55 173.930 3.837 0.584022 
7 203 4 Ol 26 202. 857 5.902 0.771020 
8 232 4 35 56 231.731 8.598 0.934889 
9 261 5 10 26 260.530 12.006 1.079413 
10 290 5 44 57 289.224 16.209 1.209768 
11 319 6 19 29 317.776 21.287 1.328110 
12 348 6 54 Ol 346. 142 27.316 1.436420 
13 377 7 28 34 374.271 34.373 1.536216 
14 406 8 03 07 402.100 42.528 1.628679 

8 37 40 : 

ce. CHORD-LENGTH = 30 

1 30 0° 33’ 20” 30.000 0.0486 8.639817 
2 60 1 06 40 59.999 -2182 9.338785 
3 90 1 40 00 89.997 .6108 9.785937 
4 120 2 13 20 119.989 1.309 0.116914 
5 150 2 46 41 149.969 2.400 0.380123 
6 180 3 20 02 179.928 3.970 0.598746 
7 210 8 53 22 209 . 852 6.106 0.785748 
8 240 4 26 44 239.122 8.894 0.949112 
9 270 5 00 05 269.514 12.420 1.094137 
10 300 5 33° 27 299.197 16.768 1.224491 
ll 330 6 06 49 328.734 22.021 1.342833 
12 360 6 40 12 358.078 28.258 1.451144 
13 390 7 13 36 387.176 35.558 1.550939 
14 420 vf Bt y 415.965 43.995 1.643402 

8 2 























AND COORDINATES XY AND Y 


c. CHORD-LENGTH = 31 


en LS 








259 























n nc Ds y x Log «x 
1 31 0° 32’ 15” 31.000 0.0451 8: 654058 
9 62 1 04 31 61.999 2254 9.353026 
3 93 1 36 47 92.997 "6312 9.800177 
4 124 2 09 02 123.988 1.353 0.131154 
5 155 2 41 18 154.968 2.479 0.394363 
6 186 3 13 34 185.925 4.102 ” 0.612986 
7 217 3 45 50 216.84" 6.309 0.799984 
8 248 4 18 07 247.713 9.191 0.963353 
9 279 4 50 24 278.498 12.834 1.108377 
10 310 5 22 41 309.170 17.327 1.238732 
1 341 5 54 59 339.692 22.155 1.357073 
12 372 6 27 17 370.014 29 200 1465384 
13 403 6 59 35 400. 082 36.743 1.565179 
7 31 53 
eee Cae er ee ee eee 
+. ¢ CHORD-LENGTH = 32 
1 32 0° 31’ 15” 32.000 0.0465 8.667846 
2 64 1 02 30 63.999 9307 9.366814 
3 96- | 1 33 45 95.997 6516 9.813965 
4 128 2 05 00 127.988 1.396 0.144942 
5 160 2 36 16 159.967 2.559 0.408152 
6 192 3 07 31 191.923 4.234 0.626774 
7 904 3 38 47 993 842 6.513 0.813772 
8 256 4 10 03 256 . 703 9.487 0.977141 
9 998 4 41 19 287.48] 13.248 1.122165 
“10 320 5 12 36 319.144 17.886 1.252520 
ul 352 5 43 53 350.649 23, 489 1.370862 
12 384 6 15 10 38] 950 30.142 1.479172 
13 416 6 46 28 412.988 37.929 1.578968 
7 17 46 
c. CHORD-LENGTH = 33 
1 33 0° 30’ 19” 33.000 0.0480 8.681210 
2 66 1 00 36 65.999 2400 9.380178 
3 99 1 30 55 98.997 ‘6719 9. 827329 
4 132 2 01 1B 131.988 1.440 0.158306 
5 165 2 31 32 164.966 2.639 0.421516 
6 198 3 Ol 50 197.921 4.367 0.640138 
7 23] 3 32 09 230. 837 6.716 0.827136 
8 264 4 02 28 263.694 9.784 0.990505 
9 297 4 32 48 296.465 13.662 1.135529 
10 330 5 03 07 329.117 18.445 1.265884 
ia 363 5 33 27 361.607 24.223 1.384226 
12 396 6 03 47 393.886 31.084 1492536 
13 429 6 34 01 425 872 39.114 1.592332 


260 TABLE XXXVI.—DEGREES OF CURVES 


c. CHORD-LENGTH = 34 





n nc Ds ¥ a Log x 
|} |] AA] |] 
1 34 Oreos 2p" 34.000 0.0495 8.694175 
2 68 0 58 49 67.999 2473 9.393143 
3 102 1 28 14 191.996 6923 9.840294 
4 136 1 57 39 135.987 1.483 0.171271 
5 -170 2 et 04 169.965 2.719 0.434481 
6 204 2 56 29 203.918 4.499 0.653103 
7 238 Secor OD 287 . 832 6.920 0.840101 
8 272 3 55 20 271.685 10.080 1.003470 
9 306 4 24 46 305.449 14.076 1.148494 
10 340 ARE OAtaL 2 339.090 19.004 1.278849 
11 374 5123838 372.565 24.957 1.397191 
12 408 oe He 405. 822 32.026 1.505501 





c. CHORD-LENGTH = 35 





1 35 0° 28’ 34’ 35.000 0.0509 8.706764 
2 70 0 57 09 69.999 . 2045 9. 405732 
3 105 1 25 43 104.996 7127 9. 852883 
a 140 1 54 17 139 . 987 1.527 0.183860 
5 175 2 e2 52 174.964 2.799 0.447070 
6 210 2 51 27 209.916 4.631 0.665692 
7 245 3 20 Ol 244 . 827 7.123 0.852690 
8 280 3 48 36 279. 675 10.377 1.016059 
9 315 4 17 12 314. 433 14.490 1.161083 
10 350 4 45 47 349. 063 19.563 1.291488 
1] 385 5 14 23 883. 523 25.691 1.409780 
12 420 : a se 417.758 32.968 1.518090 


c. CHORD-LENGTH = 386 











1 36 0° 27’ 47” 36.000 0.0524 8.718998 
2 72 0 55 33 71.999 2618 9.417967 
3 108 1 238 20 107.996 . 7330 9.865118 
4 144 1 51 07 143. 987 1.571 0.196095 
5 180 2 18 54 179.963 2.879 0. 459304 
6 216 2 46 41 215.913 4.764 0.677927 
7 202 3 14 28 251.822 7.327 0.864924 
8 288 3 42 15 - 287. 666 10.673 1.028293 
9 324 4 10 08 323.417 14.905 1.173318 
10 360 4 37 51 359 . 037 20.122 1.303673 
1] 396 5 05 39 394. 480 26.425 1.422014 
12 432 : a - 429.694 33.910 1.530325 


AND COORDINATES XY AND Y 


C. 


CHORD-LENGTH = 87 


261 




















n nec Ds y x Log x 
1 37 Oeezi 024 37.000 0.0538 8.730898 
2 74 0 54 08 73.999 .2691 9.429866 
3 lll P21, 05 110.996 7534 9.877017 
4 148 1 48 07 147.986 1.614 0.207994 
5 185 2 15 09 184.962 2.959 0.471203 
6 222 2 42 11 221.911 4.896 0.689826 
7 259 3 09 13 258. 817 7.530 0.876824 
8 296 3 386 15 295. 657 10.970 1.040193 
9 333 4 03 17 332.400 15.319 1.185217 
10 370 4 30 20 369.010 20.681 1.315572 
ll 407 A 5T 23 405. 488 27.159 1.433913 
2) ict..26 
c. CHORD-LENGTH = 38 
hc ch 
1 38 0° 26’ 19’ 38.000 0.0553 8.742480 
2 76 - 0 52 39 75.999 .2163 9.441448 
3 114 TSS) 57 113.996 71737 9.888599 
4 152 Ay 16 151.986 1.658 0.219576 
5 190 Gale 35 189.961 3.039 0.482785 
6 228 2 87 54 227.909 5.028 0.701408 
y 266 3 04 14 265.812 7.734 0.888406 
8 804 3. 30 33 303 . 648 11.266 1.051774 
9 342 3 56 53 341.384 15.733 1.196799 
10 3380 4 23 13 378. 983 21.240 1.327154 
ll 418 4 49 33 416.396 27.893 1.445495 
Gy hay alas! 
c. CHORD-LENGTH = 39 
1 39 Opcbeokm 39.000 0.0567 8.753761 
2 78 OF bie 17 77.999 . 2836 9.452729 
3 117 PeelG 55 116.996 7941 9.899880 
4 156 1) 42: 34 155.985 1.702 0.220857 
5 195 2 08 18 194.960 3.119 0.494066. 
ae 234 2 33 51 233.906 5.160 0.712689 
7 273 2 59 30 272.807 7.938 0.899687 
8 312 3-25 09 311: 638 11.563 1.063055 
9. 351 3 50 48 850.368 16.147 1.208080 
10 390 4 16 28 388. 956 21.799 1.338435 
ll 429 4 42 07 427.354 28.627 1.456776 
5 O7 46 





262 

















n ne Ds y 3 Log x 
] 40 0° 25’ 00” 40.000 0.0582 8.764756 
2 80 0 50 00 79.999 . 2909 9.463724 
3 120 155500 119.996 8145 9.910875 
4 160 1 40 00 159.985 1.745 0.241852 
5 200 2 05 00 199. 959 3.199 0.505062 
6 240 2 30 Ol 239. 904 5.293 0.723684 
7 280 AmepOwOL 279 . 802 8.141 0.910682 
8 320 3 20 01 319. 629 11.859 1.074051 
9 360 38 45 02 359 . 852 16.561 1.219075 
10 400 Are O03 398.929 22.358 1.349430 
ll 440 4 35 03 438.312 29.361 1.467772 
5 00 04 
ec. CHORD-LENGTH = 41 
1 41 0° 24’ 24’’ 41.000 0.0596 8.775480 
2 82 0 48 47 81.999 . 2982 9.474448 
3 123 Ee la) 10 122.996 8348 9.921599 
4 164 1 (37-34 163.985 1.789 0.252576 
5 205 em0Le 57 204.958 3.279 0.515786 
6 246 Qancomel 245.901 5.425 0.734408 
7 287 250 145 286.797 8.345 0.921406 
8 328 3 15 09 327.620 12.156 1.084775 
9 369 DUNOUEOO 368.336 16.975 1.229799 
10 410 4 03 57 408 . 903 22.917 1.360154 
4 28 21 
c. CHORD-LENGTH = 42 
1 42 Q° 23’ 49’ 42.000 0.0611 8.785945 
2 84 0 47 37 83.999 38054 9.484913 
8 126 Pulk 26 125.996 8552 9.932065 
4 168 1+ 35. 14 167.984 iesse 0.263042 
5 210 1 59 02 209.957 3.359 0.526251 ~ 
6 252 caree hoe 251.899 5.557 0.744874 
Ht 294 2 46 41 293 . 792 8.548 0.931871 
8 336 3 10 30 335.611 12.452 1.095240 
9 378 3 34 19 377.319 17.389 1.240265 
10 420 : a Ne 418.876 23.476 1.370619 








TABLE 











XXX VI.—DEGREES 


c. 


OF CURVES 


CHORD-LENGTH = 40 





























AND COORDINATES Y AND Y 


ec. CHORD-LENGTH = 43 


263 



























































n ne ° Ds y x Log x 
1 ey US eae ae 43.000 0.0625 8.796164 
2 86 0 46 31 85.999 8127 9.495133 
3 129 1 09 46 128.996 .8755 9.942284 
4 172 1 33 02 171.984 1.876 0.273261 
5 215 dee bG 17 214.955 3.439 0.536470 
6 258 2 19 33 257 . 897 5.690 0.755093 
7 301 2 42 48 300. '787 8.752 0.942090 
8 344 3 06 04 343. 601 12.749 1.105459 
9 387 3 29 20 386 . 303 17.803 1.250484 
10 43) 31524) 135 428.849 24.035 1.380839 
4 15 50 
c. CHORD-LENGTH = 44 
1 44 . We pee ial 44.000 0.0640 8.806149 
2 88 Ome45r 2 87.999 .38200 9.505117 
3 132 1 08 ll 131.995 8959 9.952268 
4 176 1pa0 05 175.984 1.920 0.283245 
5 220 1 53 38 219.954 3.519 0.546454 
6 264 2° 16 22 263. 894 5.822 0.765077 
7 308 2 39 06 307.782 8.955 0.952075 
8 352 3 Ol 50 351.592 13.045 1.115444 
9 396 3 24 34 395, 287 18.217 1.260468 
10 440 3 47 18 438 . 822 24.594 1.390823 
4 10 02 
c. CHORD-LENGTH = 45 
1 45 O07 cemtow 45.000 0.0655 8.815908 
2 90 0 44 27 89.999 ,oete 9.514877 
3 135 1 06 40 184.995 9163 9.962028 
4 180 1 28 53 179.983 1.963 0.293005 
5 225 1 51 07 224.953 3.599 0.556214 
6 270 ce lonnu 269. 892 5.954 0.774837 
7 315 2 35.34 314.778 9.159 0.961834 
8 360 2 57 48 359.583 13.341 1.125203 
9 405 3 20 Ol 404.271 18.631 1.270228 
10 450 31.4215 448 . 796 25.153 1.400583 
4 04 2 
c, CHORD-LENGTH = 46 
st 46 UE DAS Iie i ee 46.000 0.0669 8.825454 
2 92 0 43 29 91.999 .8345 9.524422 
3 138 } 05 13 137.995 .9366 9.971573 
4 184 1 26 58 183. 983 2.007 0.302550 
5 230 1 48 42 229.952 3.679 0.565759 
6 276 2 10 26 275.889 6.087 0.784382 
ty 822 2fi3e7 11 321.773 9.362 0.971380 
8 368 2ardS noo 367.573 13.638 1.134749 
9 414 & 15 40 413.255 19.045 1.279773 
10 460 > as a 458.769 25.712 1.410128 


264 











TABLE XXXVI.—DEGREES 








OF CURVES 


AND COORDINATES XY AND Y 


Cc. 


CHORD-LENGTH = 47 






































n ne Ds Yy av Log zx 
l AY OMe a6” 47.000 0.0684 8.834794 
2 94 0 42 33 93.999 .8418 9.533762 
3 141 E03: 250 140.995 .9570 9.980913 
4 188 255.06 187.982 2.051 0.311890 
5 235 1 46 23 284.951 3.759 0.575100 
6 282 2 O07 40 281. 887 6.219 0.793722 
7 329 2 28 57 328.768 9.566 0.980720 
8 376 2 50 14 875.564 13.934 1.144089 
9 423 Seceh Spl 422.238 19.459 1.289113 
10 470 3 382 48 468.742 26.270 1.419468 
3 54 05 
c. CHORD-LENGTH = 48 
1 48 O20 50” 48.000 0.0698 8.848937 
2 96 0 41 40 95.999 .8491 9.542905 
3 144 1025 30 143.995 9774 9.990057 
4 192 1 23 20 191.982 2.094 0.321034 
5 240 1 44 10 239 . 950 3.839 0.584243 
6 288 2 05 00 287.885 6.351 0.802866 
7 336 225-51 335.763 9.769 0.989863 
8 384 2 46 41 383.555 14.231 1.153232 
9 432 a OT saL 431.222 19.873 1.298256 
DENCOmAL 
c. CHORD-LENGTH = 49 
1 49 0°320/- 25” 49.000 0.0713 8.852892 
cue 98 0 40 49 97.999 . 3563 9.551860 
3 147 1 O1 14 146.995 .9977 9.999011 
4 196 e215 38 195.982 2.188 0.329988 
5 245 1 42 -03 244.949 3.919 0.593198 
6 294 oenOoe OT 298 . 882 6.484 0.811820 
7 343 2. eeeDe 342.758 9.973 0.998818 
8 392 2 43 17 391.546 14.527 1.162187 
9 44] 3 03 41 440.206 20.287 1.307211 
3 24 06 
ce CHORD-LENGTH = 50 
1 50 0° 20’ 00” 50.000 0.0727 8.861666 
2 100 0 40 00 99.999 .3636 9.560634 
3 150 1 00 00 149.995 1.018 0.007785 
4 200 1 20 00 199.981 2.182 0.338762 
5 250 1 40 00 249 . 948 3.999 0.601972 
6 300 2 00 00 299.880 6.616 0.820594 
7 350 2 20 00 349.753 10.176 1.007592 
8 400 2 40 Ol 399.536 14.824 1.170961 
9 450 : a A 449.190 20.701 1.315985 


sf 
{ 


| 


_ 
SCHOND OPWOe s 


—" 
coy) 


TABLE XXXVII.—FUNCTIONS OF SPIRAL ANGLE § 























cos s | log vers s 
0° 10’ 99999 | 4.626422 
0 30 99996 | 5.580662 
1 00 99985 | 6.182714 
1 40 99958 | 6.626392 
2 30 99905 | 6.978536 
3 30-| .99813 | 7.270726 
4 .40 99668 | 7.520498 
6 00 99452 | 7.738630 
7 30 99144 | 7.982227 
9 10 98723 | 8.106221 
11 00 98163 | 8.264176 
13 00 97437 | 8.408748 
ay SL 96517 | 8.541968 
17 30 95372 | 8.665422 
20 00 93969 | 8.780370 
22 40 92276 | 8.887829 
25 30 | .90259 | 8.988625 
28 30 87882 | 9.083441 
31 40 85112 | 9.172846 
35 00 81915 | 9.257814 








265 








TPP Oe : ; 

VeES § sin s log sin s 8 
-024 || .00291 7.463726 On 10: 
-218 || .00873 7.940842 0 30 
.873 || .01745 | 8.241855 1 00 

2.424 02908 | 8.463665 1 40 
5.453 04862 | 8.639680 2 30 
10.687 || .06105 | 8.785675 3 30 
18.994 | .08186 | 8.910404 4 40 
31.388 || .10458 | 9.019235 6 00 
49.018 | .18053 | 9.115698 7 30 
73.173 || .15981 | 9.202284 9 10 

105.270 19081 | 9.280599 11 00 

146.857 22495 | 9.352088 13 00 

199.570 26163 | 9.417684 ars age 

265.186 30071 | 9.478142 17 30 

345.540 34202 | 9.534052 20 00 

442 543 38537 | 9.585877 22 40 

558.153 43051 | 9.683984 25 «30 

694.335 AT716 | 9.678663 28 30 

853. 050: 52498 | 9.720140 31 40 

1036.20 57358 | 9.758591 385 00 

















— 266 TABLE XXXVIII.—SPIRAL TANGENTS, 


Notr.—These Coordinates locate the point A’ where the Spiral arc 
(n + 1) produced backward would meet a parallel tangent. They do not 
apply to the selected curve unless this has the same Degree of curve as 
the Spiral arc. (See Table XX XIX.) 


CHORD, 10 FEET 
























































Tangent Tangent L. Chord Offset Dist. 
n SE. LE. SL. D. q. 
3 18.334 11.667 30.000 .07 15.00 
4 25.001 15.001 39.999: 15 20.00 
5 31.670 18.337 49.996 sao 25.00 
6 38. 339 21.674 59.991 Al 29.99 
7 45.017 25.016 69.980 .61 34.99 
8 51.700 28.363 79.962 87 89.98 
9 58.390 81.719 89. 933 1.20 44.96 
10 65.094 35. 086 99.889 1.60 49.94 
- il 71.816 38.469 109.824 2.08 54.91 
12 78.560 41.873 119.731 2.64 59. 87 
13 85.333 45.304 129. 602 3.30 64.81 
14 92.144 48.768 189. 429 4.06 69.73 
15 99.001 52.276 149.200 4.92 74.63 
16 105.916 55. 835 158.903 5.90 79.50 
17 112.902 59. 459 168.524 6.99 84.34 
18 119.972 63.161 178.048 8.21 89.13 
19 127.144 66.956 187.457 9.56 93. 88 
20 134.439 70.863 196.731 11.04 98.58 
CHORD, 11 FEET 
Tangent Tangent L. Chord Offset Dist. 
n SE. LE. SL. Dp. q. 
3 20.17 12.83 33.00 08 16.50 
4 27.50 16.50 44.00 16 22.00 
5 34.84 20.17 55.00 28 ar. 
6 42.17 23.84 65.99 45 32.99 
7 49.52 27.52 76.98 67 38.49 
8 56.87 31.20 87.96 96 43.98 
9 64.23 34.89 98.93 1.32 49. 46 
10 71.60 38.59 109. 88 1.76 54.94 
11 79.00 42.32 120.81 2.28 60.40 
12 86.42 46.06 131.70 2.91 65.86 
13 93.87 49 . 83 142.56 3.63 71.29 
14 101.36 53.65 153.37 4.46 76.71 
15 108.90 57.50 164.12 5.41 82.09 
16 116.51 61.42 174.79 6.49 87.45 
17 124.19 65.41 185.38 7.69 92.77 
18 131.97 69.48 195.85 9.03 98.05 
19 139.86 73.65 206.20 10.51 103.27 
20 147.88 77.95 216.40 12.14 108. 43 
eee ee Ae, eee) eS eee 


LONG CHORD AND COORDINATES, p AND gq 267 
CHORD, 12 FEET 


2 




















CHORD, 18 FEET 








Tangent Tangent L. Chord Offset Dist. 
SE. LE. SL. p. q. 
22.00 14.00 36.00 09 18.00 
30.00 18.00 48.00 ny 24.00 
38. 00 22.00 60.00 .3l 30.00 
46.01 26.01 71.99 .49 35.99 
54.02 30.02 83.98 5(p3 41.99 
62.04 34.04 95.95 1.05 47.98 
70.07 38.06 107.92 1.44 53.96 
78.11 42.10 119.87 1.92 59.93 
86.18 46.16 131.79 2.49 65.90 
94.27 50.25 143.68 DaLT 71.84 

102.40 54.36 155. 52 3.96 UB he hf 
110.57 58.52 167.31 4.87 83.68 
118.80 62.73 179.04 5.90 89.56 
127.10 67.00 190.68 7.08 95.40 
135.48 71.35 202.23 8.39 101.21 
143.97 75.79 213.66 9.85. 106.96 
152.57 80.35 224.95 11.47 112.66 
161.33 85.04 236.08 13.24 118.29 


Tangent Tangent L. Chord Offset Dist. 
SE. LE. SL. p. q. 
23.83 15.17 39.00 09 19.50 
32.50 19:50 52.00 19 26.00 
41.17 23.84 64.99 03 32.50 
49.84 28.18 77.99 03 38.99 
58.52 82.52 90.97 -19 45.49 
67.21 36.87 103.95 1.18 51.97 
75.91 41.23 116.91 1.56 58.45 
84.62 45.61 129.86 2.08 64.93 
93.36 50.01 142.77 2.70 71.39 

102.13 54.43 155.65 3.43 77.83 
110.93 58.89 168.48 4.29 84.25 © 
119.79 63.40 181.26 5.27 90.65 
128.70 67.96 193.96 6.40 97.02 
137.69 72.59 206.57 VEG 103.35 
146.77 B30 219.0 9.09 109.64 
155.96 82.11 231.46 10.67 115.87 
165.29 87.04 243.69 12.42 122.05 
174.77 92.12 255.75 14.35 128.15 





268 








TABLE XXXVIII.—SPIRAL TANGENTS, 





CHORD, 14 FEET 























Tangent Tangent L. Chord Offset Dist 
Ue SE. LE. SL. D. q. 
3 25.67 16.33 42.00 10 21.00 
4 35.00 21.00 56.00 .20 28.00 
4 44.34 25.67 69.99 .36 35.00 
6 53.67 30.34 83.99 Bae 41.99 
ih 63.02 35.02 97.97 .86 48.98 
8 72.38 39.71 111.95 l 22 55.97 
9 81.75 44.4] 125.91 1.68 62.95 
10 91.13 49.12 139. 84 Piya 69.92 
11 100.54 53. 86 153.75 2.91 76.88 
12 109.98 58.62 167.62 3.70 * 83.82 
13 119. 47 63.43 181.44 4.62 90.74 
14 129.00 68.28 195.20 5.68 97.63 
15 138.60 73.19 208.88 6.89 104.48 
16 148. 28 78.17 222.46 8.26 111.30 
17 158.06 83.24 235.93 9.79 118.07 
18 167.96 88. 48 249.27 11.49 124.79 
19 178.00 93.74 262.44 13.38 131. 44 
20 188.21 99.21 275.42 15.45 138.01 

CHORD, 15 FEET 

‘Tangent Tangent L. Chord Offset Dist. 
ay SE. LE. SL. D. q. 
3 27.50 17.50 45.00 sib} 22.50 
4 37.50 22.50 60.00 see 30.00 
5 47.51 27.51 74.99 38 37.50 
6 57.51 32.51 89.99 .61 44.99 
i 67.53 37.52 104.97 » .92 52.48 
8 77.55 42.54 119.94 131 59.97 
9 87.59 47.58 134.90 1.80 67.45 
10 97.64 52.63 149.83 2.40 74.91 
11 107.72 57.70 164.74 3.11 82.37 
12 117.84 62.81 179.60 3.96 89.80 
13 128.00 67.96 194. 40 4.95 97.22 
14 138.22 73.15 209.14 6.09 104.66 
15 148.50 78.41 223.80 7.38 111.95 
16 158.87 83.75 238.35 8.85 119.25 
17 169.35 89.19 252.79 10.49 126.51 
18 179.96 94.74 267 . 07 12.31 133.70 
19 190.72 100.438 281.19 14.33 140.82 
20 201.66 106.29 295.10 16.55 147.86 














LONG CHORD AND COORDINATES, p AND-g 269 


CHORD, 16 FEET 











Tangent Tangent L. Chord Offset Dist. 
SE. LE. SL. D. q. 

3 29.33 18.67 48.00 12 24.00 
4 40.00 24.00 64.00 23 32.00 
5 50. 67 29.34 79.99 4] 40.00 
6 61.34 34.68 95.98 65 47.99 
7 72.03 40.03 111.97 98 55.98 
8 82.72 45.38 127.94 1.40 63.97 
9 93. 42 50.75 143.89 1.92 71.94 
10” |e 104-15 56.14 159. 82 2.56 79.91 
11 114.90 61.55 yaa 3.32 87.86 
12 125.70 67.00 191.57 4.23 95.79 
13 136.53 72.49 207.36 5.28 103.70 
14 147.48 78.03 223.09 6.49 iT ale ay: 
15 158.40 83.64 238.72 7.87 119.41 
16 169.47 89.34 254.24 9.43 127.20 
17 180.64 95.14 269. 64 11.19 134.94 
18 191.96 101.06 284.88 13.13 142.61 
19 203.43 107.13 299 .93 15.29 149.21 
20 215.10 113.38 314.77 17.65 157.72 














CHORD, 17 FEET 





Tangent Tangent L. Chord Offset Dist. 
SE. LE. SL. Dp. q. 








31.17 19.83 51.00 12 25.50 

















270 TABLE XXXVIII.—SPIRAL TANGENTS, 


CHORD 18 FEET 























Tangent Tangent L. Chord Offset Dist. 
n SE. LE. SL. D. q. 
3 33.00 21.00 54.00 13 27.00 
4 45.00 27.00 72.00 26 36.00 
5 57.01 33.01 89.99 46 45.00 
6 69.01 39.01 107.98 io 53.99 
Tei | 81.03 45.08 125.96 1.10 62.98 
8 93.06 51.05 143.93 1.57 71.96 
9 105.10 57.09 161.88 2.16 80.94 
10 realy 63.16 179.80 2.88 89.90 
11 129.27 69.24 197.68 3.14 98 . 84 
12 141.41 75.37 215.51 4.75 107.76 
13 153.60 81.55 233.28 5.94 116.66 
14 165.86 87.78 250.97 7.30 125.52 
15 178.20 94.10 268.56 8.86 134.34 
16 190.65 100.50 286.03 10.61 143.10 
17 203. 22 107.03 303.34 12.58 151.81 
18 215.95 113.69 320.49 14.78 160.44 
19 228.86 120.52 337. 42 17.20 167.99 
20 241.99 127.55 354.12 19.86 177.44 











CHORD, 19 FEET 


Tangent Tangent L. Chord Offset Dist. 
SE. LE. L 








34.83 22.17 57.00 14 28.50 
47.50 28.50 76.00 .28 38.00 
60.17 34.84 94.99 .48 47.49 


3 

4 

5 

72.84 41.18 113.98 sv 56.99 
: ' ; 



































LONG CHORD AND COORDINATES, Pp AND q 


CHORD, 20 FEET 


271 














Tangent Tangent L. Chord Offset Dist. 
SE. LE. SL. p. q. 
3 36.67 23.33 60.00 15 30.00 
4 50.00 30.00 80.00 29 40.00 
5 63.34 36.67 99.99 51 49.99 
6 76.68 43.35 119.98 gl 59.99 
7 90.04 50.03 139.96 1.22 69.98 
8 103.40 56.73 159.92 1.74 79.96 
9 116.78 63.44 179.87 2.40 89.93 
10 130.19 70.17 199.78 3.20 99.89 
11 143.63 76.94 219.65 4.15 109.83 
12 157.12 83.75 939. 46 5.28 119.74 
13 170.67 90.61 259.20 6.60 129.62 
~14 184.29 97.54 278. 86 8.11 139.47 
15 198.00 104.55 298. 40 9.84 149.26 
16 211.83 111.67 317.81 11.79 159.00 
17 295, 80 118.92 337.04 13.98 168.68 
18 939 94 126.32 356.10 16.42 178.27 
19 254. 29 133.91 374.91 19.11 186.76 
20 268.88 141.73 393. 46 22.07 197.15 
de ETC IRR Sa a oid IRN SR Ak es a 
CHORD, 21 FEET 
Tangent Tangent L. Chord Offset Dist: 
n SE. LE. SL. Dp. q. 
3 38.50 24.50 63.00 15 31.50 
4 52.50 31.50 84.00 31 42.00 
5 66.51 38.51 104.99 53 52.49 
6 80.51 45.52 125.98 86 62.99 
1 94.54 52.53 146.96 1.28 73.48 
8 108.57 59.56 167.92 1.83 83.96 
9 122.62 66.61 188.86 2.52 94.43 
10 136.70 73. 68 209.77 3.36 104.88 
11 150.81 80.79 230.63 4.36 115.32 
12 164.98 87.93 951.43 5.55 125.73 
13 179.20 95.14 272.16 6.93 136.10 
14 193.50 102.41 292. 80 8.52 146.44 
15 207.90 109.78 313.32 10.33 156.73 
16 299 42 117.25 333.70 12.38 166.95 
17 937.09 . 124.86 353.90 - 14.68 177.11 
18 951.94 132.64 373.90 17.24 187.18 
19 267.00 140.61 393. 66 90.07 196.15 
20 982. 32 148.81 413.13 93.18 207.01 
pm 


272 TABLE XXXVIII.—SPIRAL TANGENTS, 


CHORD, 22 FEET 


Tangent ‘Tangent L. Chord Offset Dist. 
SE. LE. SL p. q. 

3 40.33 25.67 66.00 16 33.00 
4 55.00 33.00 88.00 32 44.00 
5 69.67 40.34 109.99 56 54.99 
6 84.35 47.68 131.98 90 65.99 
7 99.04 55.04 153.96 1.34 76.97 
8 113.74 62.40 175.92 1.92 87.95 
9 128.46 69.78 197.85 2.64 98.92 
10 143.21 Ue Hite hy 219.76 3.52 109.88 
11 157.99 84.63 241.61 4.57 120.81 
12 172.83 92.12 263.41 5.81 131.71 
13 187.73 99.67 285.12 7.26 142.58 
14 202.72 107.29 306 .'74 8.93 153.41 
15 217.80 115.01 828.24 10.83 164.19 
16 233.02 122.84 349.59 12.97 174.90 
17 248.38 130.81 370.75 15.38 185.54 
18 263.94 138.95 391.71 18.06 196.09 
19 279.72 147.30 402.40 21.02 205.54 








CHORD, 23 FEET 














3 

4 

5 72.84 42.17 114.99 .59 57.49 
6 88.18 49.85 137.98 .94 68 . 99 
7 103.54 57.54 160.95 1.40 80.47 
8 118.91 65.24 183.91 2.01 91.95 
5) 134.30 72.95 - 206.85 2.76 103. 42 
10 149.72 80.70 229.74 3.68 114. 87 
11 165.18 88 . 48 252.59: 4.78 126.30 
12 180.69 96.31 275.38 6.08 137.70 
13 196.27 104.20 298.08 7.59 149.06 
14 211.93 112.17 320.69 9.33 160.39 
15 227.70 120.23 343.16 11.32 171.65 
16 243.61 128.42 _ 3865.48 13.56 182.85 
17 259. 67 136.76 387.61 16.08 193.98 
18 275.94 


145.27 409.50 18.88 205.01 





CHORD, 24 FEET 








42.17 26.83 69.00 at 34.50 
57.50 34.50 92.00 .33 46.00 


3 44.00 28.00 72.00 ie 36.00 — 
4 60.00 36.00 96.00 30 48.00 
4) 76.01 44.01 119.99 61 59.99 
6 92.01 52.02 143.98 . 98 71.99 
if 108.04 60.04 167.95 1.47 83.97 
8 124.08 68 . 07 191.91 2.09 95.95 
9 140.14 76.13 215.84 2.88 107.92 
10 156.23 84.21 229.73 3.84 119.86 
ll 172.36 92.33 263.58 4.98 131.79 
12 188.54 100.50 287.35 6.34 143.69 
13 204.80 108.73 311.04 7.92 155.55 
14 221.15 117.04 334.63 9.74 167.36 
15 237.60 125.46 358. 08 11.81 179.12 
16 254.20 134.01 381.37 14.15 190.80 
17 270.96 142.70 404.46 16.78 202.41 





LONG CHORD AND COORDINATES, p AND q = 273 


CHORD, 25 FEET 











Tangent Tangent L. Chord Offset Dist. 
SE. LE. SL. Dp. qa 
45.83 29.17 75.00 18 37.50 
62.50 37.50 100.00 .36 50.00 
79.18 45.84 124.99 64 62.49 
95.85 54.19 149.98 1.02 74.99 

112.54 62.54 174.95 1.53 : 87.47 
129.25 70.91 199.91 2.18 99.95 — 
145.98 79.30 224.83 3.00 112.41 
162.74 87.72 249.72 4.00 124.86 
179.54 96.17 274.56 5.19 137.28 
196.40 104.68 299.33 6.60 149.67 
213.33 113.26 324.00 8.25 162.03 
230.36 121.92 348.57 10.14 174.33 
247.50 130.69 373.00 12.30 186.58 
264.79 139.59 397.26 14.74 198.75 
282.25 148.65 421.31 17.48 210.85 











CHORD, 26 FEET 

















3 47.67 . 30.33 78.00 .19 39.00 
4 65.00 39.00 104.00 .38 52.00 
5 82.34 47.68 129.99. . 66 64.99 
6 99.68 56.35 155.98 1.06 77.98 
7 117.05 65.04 181.95 1.59 90.97 
8 134. 42 73.74 207.90 2.200 103.95 
9 151.82 82.47 233.83 3.12 116.91 
10 169.25 91.22 259.71 4.16 129.85 
11 186.72 100.02 285.54 5.40 142.77 
12 204.25 108 . 87 311.30 6.87 155.66 
13 221.87 117.79 336. 97 8.58 168.51 
14 239.57 126.80 362.51 10.55 181.31 
15 257.40 185.92 387.92 12.79 194.04 
16 275.38 145.17 413.15 15.38 206.70 














3 49.50 31.50 81.00 20 40.50 
A 67.50 40.50 108. 00 39 54.00 
5 85.51 49.51 134.99 69 67.49 
6 103.52 58.52 161.97 1.10 80.98 
ué 121.55 67.54 188.95 1.65 94.47 
8 139 .59 76.58 215.90 2.36 107.94 
9 157.65 85.64 242. 82 3.24 121.40 
10 175.75 94.73 269. 70 4.31 134.85 
11 193.90 , 103. 87 296.52 5.61 148.26 
12 212.11 113.06 323.27 7.13 161.65 
13 230.40 122.32 349.93 8.91 174.99 
14 248.79 131.67 376.46 10.95 188.28 
15° 267.30 141.14 402.84 13.29 201.51 
ee a ee es 





274 TABLE XXXVIII.—SPIRAL TANGENTS, 


CHORD, 28 FEET 





Tangent Tangent L. Chord Offset | 
SE. LE. SL. ’ p. 
3 51.33 32.67 84.00 20 | 
4 70.00 42.00 112.00 Al 
5 88.68 51.34 139.99 wal 
6 107.35 60.69 167.97 1,14 
7 126.05 70.04 195.94 ity Al 
8 144.76 79.42 223. 89 2.44 
9 163.49 88.81 251.81 3.36 
10 182.26 98.24 279.69 4.47 
ll 201.08 107.71 807.51 5.81 
12 219.97 117.24 335.25 7.40 
13 238.93 126.85 362.89 9.24 
14 258. 00 136.55 390.40 11.36 
15 277.20 146.37 417.76 13.78 














3 53.17 33.83 87.00 al 
4 72.50 43.50 116.00 42 
5 91.84 53.18 144.99 74 
6 111.18 62. 86. 173.97 1.18 
7 130.55 72.55 202.94 Lath 
8 149.93 82.25 231.89 2.53 
g 169.33 91.98 260.81 3.48 
10 188.77 101.75 289 . 68 4.63 
ll 208. 26 111.56 318.49 6.02 
12 227. 82 121.43 347. 22 7.66 
13 247.47 131.38 375.85 9.57 
14 267.22 141.43 404.34 11.77 








3 55.00 35.00 90.00 22 
a 75.00 45.00 120.00 44 
5 95.01 55.01 149.99 76 
6 115.02 65.02 179.97 1.22 
7 135.05 75.05 209.94 1.83 
8 155.10 85.09 239. 89 2.62 
9 175.17 95.16 269.80 3.60 
10 195.28 105.26 299.67 4.79 
ll 215.45 115.41 329.47 6.23 
12 230. 68 125.62 359.19 7.92 
13 256.00 135.91 388.81 9.90 
14 276.43 146.31 418.29 12.17 








LONG CHORD AND COORDINATES, p AND g = 275 


























CHORD, 31 FEET 
Tangent Tangent L. Chord Offset Dist. 
n SE. LE. SL. Dp. q. 
3 56.83 36.17 93.00 23 46.50 
4 77.50 46.50 124.00 45 62.00 
5 98.18 56.84 154.99 19 77.49 
6 118.85 67.19 185.97 1.26 92.98 
7 139.55 SES 216.94 1.89 108.46 
8 160.27 87.93 247.88 2.70 123.94 
9 181.01 98.33 278.79 3.72 139.39 
10 201.79 108.77 809. 66 4.95 154.82 
11 222.68 119.25 340.45 6.44 170.23 
12 243.53 129.81 371.16 8.19 185.60 
13 264.53 140.44 401.77 10.23 200.91 
CHORD, 32 FEET 
3 58.67 37.33 96.00 20 48.00 
4 80.00 48.00 128.00 47 64.00 
5 101.35 58.68 159.99 81 79.99 
6 122.69 69.36 191.97 1.30 95.98 
7 144. 06 80.05 223.94 1.95 111.96 
8 165, 44 90.76 255. 88 2.79 127.93 
9 186. 85 101.50 287.79 3.84 143.89 
10 208. 30 112.28 319. 64 5.11 159. 82 
11 229.81 123.10 351. 44 6.65 175.72 
12 251.39 133.99 383.14 8.45 191.58 
13 273.07 144.97 414.73 10.56. 207.39 
CHORD, 33 FEET 
3 60.50 38.50 99.00 24 49.50 
4 82.50 49.50 132.00 48 66.00 
5 104.51 60.51 164.99 84 82.49 
6 126.52 71.53 197.97 1.34 98.98 
7 148.56 82.55 230.93 2.02 115.46 
8 170.61 93.60 263.88 2.88 131.93 
9 192.69 104.67 296.78 3.96 148.38 
10 214.81 115.78 329.63 5.27 164.81 
11 236.99 126.95 362.42 6.85 181.21 
12 259.25 138.18 395.11 8.72 197.57 
13 281.58 149.50 427.69 10.89 213.87 





276 TABLE XXXVIII.—SPIRAL TANGENTS, 


CHORD, 34 FEET 
a 








Tangent Tangent L. Chord Offset Dist. 
nm SE. LE. SL. D. q. 
3 62.33 39.67 102.00 20 51.00 
4 85.00 51.00 136.00 .49 68.00 
Is 107.68 62.34 169.99 87 84.99 
6 130.35 73.69 203.97 1.38 101.98 
i 153. 06 85.05 237.93 2.08 118.96 
8 175.78 96.44 271.87 2.97 135.93 
9 198.53 107.84 305.77 4.08 152.88 
LON eee celeoe 119.29 339. 62 5.43 169.81 
11 244.17 130.80 3738.40 7.06 186.70 
12 267.10 142.37 407.08 8.98 203.56 


ce 


CHORD, 35 FEET 








3 64.17 40.83 105.00 20 52.50 
4 87.50 52.50 140.00 .51 70.00 
5 110.85 64.18 174.99 .89 87.49 
6 134.19 75. 86 209. 97 1.43 104.98 
7 157.56 87.56 244.93 2.14 122. 46 
8: 180.95 99.27 279. 87 3.05 139. 93 
9 204.37 111.02 314.77 4.20 157. 38 
10 227.83 122.80 349.61 5.59 174. 80 
1] 251.35 134.64 384. 38 7.27 - 192.19 
12 274.96 146.56 419.06 9.24 209.54 


CHORD, 386 FEET 











3 66.00 42.00 108.00 26 54.00 
4 90.01 54.00 144.00 52 72.00 
5 114.01 66.01 179.99 92 89.99 
6 138.02 78.03 215.96 1.47 107.98 
t 162.06 90.06 251.93 2.20 125. 96 
8 186.12 102.11 287. 86 3.14 143. 93 

210.21 114.19 323.76 4.32 161. 87 
10 234.34 126.31 359. 60 5.75 179. 80 
11 258.54 138.49 395.36 7.48 197.69 
12 282.81 150.74 431.03 9.51 215.53 

ee ee ee 





LONG CHORD AND COORDINATES, p AND gq = 277 


























CHORD, 37 FEET 
Tangent Tangent L. Chord Offset 
n SE. LE. SL. D. 
3 67.83 43.17 111.00 27 
4 92.51 55.50 148. 00 54 
5 117.18 67.85 184.99 94 
6 141.86 80. 20 221.97 1.51 
” 166.56 92.56 258. 93 2.26 
8 191.29 104.94 295. 86 3.23 
9 216.04 117.36 332.75 4.44 
10 | -_ 240.85 129. 82 369.59 5.91 
11 265.72 142.34 406.35 7.68 
CHORD, 38 FEET 
3 69.67 44.33 114.00 .28 
4 95.01 57.00 151.99 55 
5 120.35 69.68 189.98 .97 
6 145.69 82.36 227.96 1.55 
7 171.07 95.06 265.92 2.32 
8 196. 46 107.78 303.86 3.31 
9 991 88 120.53 341.75 4.56 
10 247.36 133.33 379.58 6.07 
ll 272.90 146.18 417.33 7.89 
CHORD, 39 FEET 
3 71.50 45.50 117.00 .28 
4 97.51 58.50 155.99 57 
5 123.51 71.51 194.98 -99 
6 149.52 84.53 233.96 1.59 
7 175.57 97.56 972.92 2.38 
8 201.63 110.62 311.85 3.40 
9 227.72 123.70 350.74 4.68 
10 253. 87 136. 84 389.57 6.23 
ll 280.08 150.03 428.31 8.09 





58.50 
78.00 
97.49 


116.98 
136.46 
155.92 
175.36 
194.78 


214.16 


278 


TABLE XXXVIII.—SPIRAL TANGENTS, 
































CHORD, 40 FEET 
Tangent Tangent L. Chord Offset Dist 
te SE. LE. SL. p. q- 

3 73.33 46.67 120.00 29 60.00 
4 100.01 60.01 159.99 .58 80.00 
5 126.68 73.35 199.98 1.02 99.99 
6 153.36 86.70 239.96 1.63 119.98 
7 180.07 100.06 279.92 2.44 139.95 
8 206.80 113.45 319.85 3.49 159. 92 
9 233.56 126.88 859.73 4.80 179.86 
10 260.38 140:34 399.56 6.39 199.77 
219.65 

CHORD, 41 FEET 

| 

3 75.17 47.83 123.00 30 61.50 
4 102.51 61.51 163.99 .60 82.00 
5 129.85 75.18 204.98 1.04 102.49 
6 157.19 88.87 245.96 1.67 122.98 
7 184.57 102.57 286.92 2.50 143. 45 
8 211.97 116.29 327.85 3.58 163.91 
9 239.40 130.05 368.73 4.92 184.36 
10 266.89 143.85 409.54 6.55 204.77 

CHORD, 42 FEET 
3 {HI 00 49.00 126.00 101 63.00 
4 105.01 63.01 167.99 61 84.00 
5 133.02 77.01 209.98 1207 104.99 
6 161.02 91.03 251.96 1.71 125.98 
if 189.07 105.07 293.92 2.56 146.95 
8 217.14 119.13 335.84 3.66 167.91 
9 245.24 133.22 377.72 5.04 188.85 
10 2738.40 147.36 419.53 6.71 209.76 








LONG CHORD AND COORDINATES, p AND gq 279 


CHORD, 43 FEET 











Comms Ow 


_ 


ro 
| OoOonte Orc 





_ 
| OwmonsS OP 


Sooo OPcw 


— 








Tangent Tangent L. Chord Offset Dist. 
SE. LE. SL. Dp. q. 
78.83 50.17 129.00 31 64.50 

107.51 64.51 171.99 . 63 86.00 
136.18 78.85 214,98 1.09 107.49 
164786 93.20 257.96 1375 128.97 
193.58 107.57 300.91 2.63 150.45 
2ee-OL 121.96 343. 84 3.75 171.91 
251.08 136.39 386.71 5.16 193.35 
279.91 150.87 429.52 6.87 214.76 





80.67 51.33 132.00 82 66.00 
110.01 66.01 175.99 64 88.00 
139.35 80.68 219.98 1.12 109.99 
168. 69 95.37 263 . 96 ig 131.97 
198. 08 110.07 307.01 2.69 153.95 
221.48 124.80 351.83 3.84 175.91 
256. 92 189.56 395.71 5.28 197.84 
286. 42 154.38 439.51 7.03 219.75 











CHORD, 45 FEET 


82.50 52.50 135.00 .33 67.50 
112.51 67.51 179.99 .65 90.00 
142.52 82.52 224.98 1.15 112.49 
172.53 97.54 269. 96 1.83 134.97 
202.58 112.57 314.91 2.75 157.45 
232.65 127.63 359. 83 3.93 179.91 
262.76 142.78 404.70 5.40 202.34 
292.92 157.89 449.50 7.19 224.74 





CHORD, 46 FEET 


84.33 53.67 138.00 .33 69.00 
115.01 69.01 183.99 .67 91.99 
145.68 84.35 229.98 1.17 114.99 
176.36 99.70 275.96 1.87 137.97 
207.08 .. 115.07 321.91 2.81 160.95 
237 . 82 130.47 367.83 4.01 183.90 
268 . 60 145.91 413.69 5.52 206. 84 
299. 43 161.40 459.49 7.35 229.74 





280 TABLE XXXVIII.—SPIRAL TANGENTS, LONG 
CHORD AND COORDINATES, p AND g 


CHORD, 47 FEET 


Tangent Tangent L. Chord Offset Dist. 
m SE. LE. SL. Dp. q- 
3 86.17 54.84 141.00 34 70.50 
4 117.51 70.51 187.99 68 93.99 
5 148.85 86.18 234.98 120 117.49 
6 180.19 101.87 281.96 1.9] 140.97 
7 211.58 117.58 328.91 2.87 164.45 
8 242.99 133.31 375.82 4.10 187.90 
9 274.43 149.08 422.69 5.64 211.33 
10 305.93 164.90 469.48 (fersy 233.73 

CHORD, 48 FEET 

3 88.00 56.00 144.00 .30 72.00 
4 120.01 72.01 191.99 70 95.99 
5 152.02 88.02 239.98 lege 119.99 
6 184. 03 104.04 287.95 1.95 143.97 
7 216.08 120.08 335.90 2.93 167.94 
8 248.16 136.14 383. 82 4.19 191.90 
9 280.27 152.25 431.68 5.76 215.83 





CHORD, 49 FEET 
a a et ee ee 


3 89.83 57.17 147.00 36 73.50 
4 122.51 73.51 195.99 71 97.99 
5 155.18 89.85 244.98 1.25 122.49 
6 187.86 106.20 293.95 2.00 146.97 
7 220.58 122.58 342.90 2.99 171.44 
8 253.33 138.98 391.82 4.27 195.90 
9 286.11 155.42 440.68 5.88 220.33 


CHORD, 50 FEET 


91.67 58.34 150.00 00) 4 75.00 
125.01 75.01 199.99 73 99.99 
158.35 91.68 249.98 1.27 124.99 


3 

4 

5 

6 191.70 108.37 299.95 2.04 149.97 
7 225.09 125.08 349. 90 3.05 174.94 
8 208. 50 141.82 399.81 4.36 199.90 
9 291.95 158.59 449. 67 6.00 224.82 





TABLE XXXIX.—SELECTED CURVES WITH 281 
PROPER SPIRALS 
































Ds ie 8 R’ vers s | R’ sins q Dp 
1° 40° 3 40 1 S500' 0.524 59.999 59.987 0.291 
1 40 4 50 1 49 1.454 99.989 99.992 0.727 
2 3 33 1 00 0.436 50.000 48.997 0.236 
2 4 42 1 40 1.212 83.326 84.658 0.620 
2 5eep0 2 30 2.727 124.967 124.981 1.272 
2 30 Bh ual 1 00 0.349 40.001 40.996 0.201 
2 30 4 33 1 40 0.970 66. 663 65.325 0.470 
2 30 5 40 2° 30 2.181 99.976 99.983 1.018 
2 30 6 AT 3 30 4.275 139.924 || 141.968 1.944 
3: Stee 1 00 0.291 33.335 32.663 0.157 
3 4 28 1 -40 0.808 55.554 56.436 0.414 
3 5 33 2 30 1.818 83.317 81.649 0.821 
3 6 38 3 30 3.563 116.607 111.302 1.465 
3 7 44 4 40 6.332 155.401 152.381 2.623 
3 8 50 6 00 10.464 199.658 199.878 4.360 
3 20 3 20 1 00 0.262 30.003 29.995 0.145 
3 20 Agee) 1 40 0.727 50.000 49.991 0.364 
3 20 5 30 2 30 1.636 74.987 74.982 - 0.764 
3 20 6 35 3 30 3.207 104.950 104.966 1.424 
3 20 7 40 4 40 5.699 139. 865 139.937 2.442 
3 20 8 45 6 00 9.417 179.697 179.885 3.924 
3 20 ROU 7 30 14.707 224.390 224.800 5.994 
4 4°21}. .1 ~40 0.606 41.669 42 323 0.310 
4 Li as 2 30 1.364 62.493 62.481 0. 636 

4 6 29 3 30 2.672 87.463 86. 467 1.165 
4 (ous) ASAAQ a ANT5O 116.561 |) 114.276 1.966 
4 8 37 6 00 7.848 149.757 145.900 3.122 
4 9 41 7 30 12.257 187.003 181.333 4.718 
4 10 46 9 10 18.297 228. 237 230.532 7.415 
4 10 4 20 1 40 0.582 40.003 39.990 0.291 
4 10 5 24 2 30 1.309 59.994 59.981 0.611 
4A 10 6 28 3 30 2.565 83.966 83.967 1.140 
4 10 7 132 4 40 4.560 111.901 111.941 1.953 
4 10 8 36 6 00 7.535 143.769 143.897 3.138 
Aaa ott. 9 AD 7 30 11.867 179.526 179.826 4.694 
4 10 10 44 9 10 17.565 219.111 219.711 7.029 
5 5 20 2 30 1.091 50.000 49.979 0.509 
5 6 23 3 30 2.138 69.979 67.966 0.905 
5 Lek 4 40 3.800 93.260 95. 606 1.695 
5 8 30 6 00 6.279 119.819 119.903 2.615 
5 9 33 7 30 9.807 | 149.619 146. 846 3.855 
5 10 37 9 10 14.639 182.610 186.400 6.042 
5 ll 40 11 00 21.060 218.720 219.592 8.301 
5 20 ve 2 30 1.023 46.877 48.103 0.497 
5eee0 Ge 22 hese a0 2.004 65.608 66.3839 0.907 
5 20 Teco ve a 40 3.563 87.435 87.441 1.525 
520 8 28 6 00 5.887 112.335 111.405 2.414 
§.E20 9 31 7 30 9.408 140.275 138. 223 3.426 
5 20 10 34 9 10 13.725 171.204 167.886 5.279 
i PAD) herby ll 00 19.745 205.060 200.378 7.414 
Bb0) 5 17 2 30 0.935 42.862 42.120 0.425 
SOUL he 162220 3 30 1.833 59.988 59.964 0.813 
5 50 7 23 4 40 3.258 79.946 80.940 1.423 











282 





AAOMWPArRWrca Ooo 


S2 SO? S32 G2 O32 OS? C2 


WOOWOOOOOO 00000000000000 60000000000000 aaa gaQ 9-9-3 











TABLE XXXIX.—SELECTED CURVES WITH 


— 
MmOWOCORIOO HOC 3 


— 








I he = i 
OID Wr OI POO WHOS Po? HOW hwR HO 


id 
>) Te) 


dl 
CUD IO 





R’ vers s 


5.383 





R’ sin s 


102.714 
128.260 
156.541 
187.496 


41.673 
58.324 
17.727 
99. 863 
124.700 
152.196 
182.293 


52.497 
69.962 
89.886 
112.242 
136.991 
164.081 
193.440 


50.000 
66 . 634 
85.611 
106.904 
130.476 
156.278 
184.240 


62.198 
79.911 
99.786 
121.788 
145.871 
171.976 
200.011 


58.316 
74.924 
93.558 
114. 188 
136.768 
161.240 
187.529 


55.988 
71.932 
89 . 822 
109. 628 
131.306 
154.801 
180.041 


51.848 
66.613 
83.181 
101.523 
121.598 
143.356 
166.729 
191.632 











105.045 
132.270 
152.629 
185.069 


43.309 
55.630 
76.164 
99.905 
126.846 
156.974 
179.314 


55.460 
69.939 
93.901 
112.353 
142.260 
164.653 
200.446 


51.959 
66.272 
82.194 
108.707 
128. 828 
161.498 
185.774 


63.713 
79.904 
97.857 
117.570 
149.989 
174. 166 
200.071 


155. 533 
181.324 


53.078 
69.229 
87.511 
97.942 
119.473 
143.107 
168.824 
196. 603 














nr 











PROPER SPIRALS 


R’ vers s. 





R’ sin s 


195.142 


62. 436 
76. 202 
91.271 
107. 603 
125.147 
143. 839 
163. 602 
184.337 


48.007 
59.948 
73.166 
87. 634 
103.315 
120.160 
138.107 
157. 082 
176.991 
197.723 


35.040 
45.019 
56.216 
68.612 
82.179 

















203.724 


55. 642 
73.528 
83. 850 
105.559 
117.712 
143.193 
157. 150 
186.330 


50.948 
68. 662 
78.903 
100.619 
112.873 
138,541 
152.772 
176.308 
198. 456 


193.517 


47. 882 
56.841 
76. 433 
87.691 
99.596 
125.052 
139. 203 
153, 980 
185.119 
202.042 


34.911 
42.879 
51.590 
71.013 


82.188 








TABLE XXXIX.—SELECTED CURVES WITH 



































D! n R’ vers s R’ sin s q p 
doeee0: 12 16 13° 00’ 11.039 96. 884. 94.091 4.082 
13 20 13 17 15 10 15.001 112.680 106.720 5.149 
13 20 14 18 17 30 19.934 129.511 120.068 6.463 
13 20 15 20 20 00 25.974 146.968 149.281 9.785 
13 20 1621 22 40 33.266 165.974 164.649 11.920 
13 20 Lied: 20 380 41.955 185.416 197.692 16.920 
14 10 8 ll 6 00 2.221 42.384 45.514 1.040 
14 10 9 12 tomo 3.469 52.925 54.881 1.499 
14 10 10 13 9 10 5.178 64.595 65.057 2.088 
14 10 ll 14 11 00 7.467 77.368 76.041 2.809 
14 10 T2e5 13 00 10.392 91.212 87.827 313% 
14 10 13 16 15 10 14.123 106.083 100.411 4.841 
14 10 14 18 17 30 18.767 121.928 127.651 7.630 
14 10 15 19 20 00 24.453 138: 680 142.757 9.518 
14 10 16 20 22 40 31.318 156.257 158.622 11.717 
14 #10 I WAC APAL rays) 39.499 174.561 175.234 14.257 
14 10 18 22 28 30 49.136 193.475 192.579 17.167 
15 8 10 6 00 2.098 40.041 39.866 0.867 
15 9 ll W 730 3.277 50.000 48.822 Wei 
15 10 12 9 10 4.892 61.025 58.654 1.815 
15 Lies ll 00 7.038 73.092 69.359 2.504 
15 124 13 00 9.818 86.171 80.982 3.369 
15 13. (15 15 10 13.343 100.220 93.368 4.436 
15 14217 17 330 17.729 115.190 120.524 7.201 
15 iby 1: 20 00 23.102 131.016 135.608 9.081 
15 16 19 22 40 29.587 147.621 151.514 11.296 
15 Vigsed 25 30 37.316 164.913 168.225 13.880 
15 18 2l 28 30 46.421 182.783 185.723 16.868 
16 40 9 10 7 30 2.951 45.030 44.808 1.189 
16 40 10;,A12 9 10 4.406 54.960 54.746 1.742 
16 40 Lie ll 00 6.338 65.827 65.666 2.470 
16 40 Zee 13 00 8.842 77.606 77.561 3.403 
16 40 13 14 15 210 12.016 90.259 90.423 4.578 
16 40 ]4° 715 17 30 15.967 103.740 104.243 6.030 
16 40 15 16 20 00 20.805 117.994 119.005 7.802 
16 40 Gch Wr 22 40 26.646 132.948 134.699 9.933 
16 40 17 18 PAs Si) 83.607 148 . 522 151.302 12.469 
16 40 18 19 28 30 41.807 164.615 168.795 15.455 
16 40 19 20 31 40 51.362 181.112 187.152 18.939 
18 20 10 10 9 10 4.008 50.000 49.732 1.581 
18 20 lI dl ll 00 5.767 59. 887 60.649 2.307 
18 20 ele 13 00 8.044 70.603 72.628 3.259 
18 20 Isls 15° J0 10.932 82.115 85.661 4.477 
18 20 14 14 1730 14.526 94.380 99.737 6.005 
18 20 15 15 20 00 18.928 107.347 114.840 7.891 
18 20 17 16 20 °30 80.575 135.120 131.390 10.382 
18 20 TQ 28 30 38. 034 149.760 148.554 13.200 
18 20 19 18 31 40 46.729 164.769 166.668 16.542 
18 20 20 19 85 00 56.761 180.023 185.702 20.465 
20 ll 10 ll 00 5.290 54.941 54.637 2.050 
20 [2ealt 13 00 7.380 64.772 66 . 523 2.981 
20 13" 12 15, 410 10.029 75.333 79.538 4.194 
20 15e-13 20 00 17.365 98.481 94.081 5.878 
20 16 14 22 40 22.240 110.962 109. 453 7.884 
20 Led 25 30 28.049 123.961 125.892 10.348 
20 18 16 28 30 84.893 137.392 143.374 13.328 
20 JOA? 31 40 42.869 151.161 161.863 16.887 
20 20 18 385 00 52.073 165.154 181.322 21.088 











PROPER SPIRALS 























D’ n ¢ 8 R’ vers s R’ sin s q Dp 
22° 30’ 12 10 13° 00’ 6.569 57.653 61.706 2.850 
22 30 14 ll 17 30 11.862 77.068 75.453 4.269 
22 30 15e Az 20 00 15.456 87.657 90. 092 5.999 
22 30 16 13 22 40 19.795 98.767 105.904 8.177 
22 30 18 14 28 30 31.058 122.264 123. 406 11.135 
22 30 19 15 31 40 38.158 134.547 141.651 14.568 
22 30 20 16 35 00 46.350 147.003 160.976 18.682 
25 14 10 17 30 10.692 69. 466 _ 69.189 3.973 
25 15 1 20 00 13.932 79.010 83.927 5.735: 
25 1% 12 25 30 22.504 101.769 98.114 8.214 
25 18 13 28 30 27.995 110.229 117.894 11.184 
25 20 14 35. 00 41.778 182.502 136.979 15.125 

e230 ss ay 20 00 12.628 71.948 76.177 5.251 
27 +30 fees PL 25 30 20.492 90.563 92.663 7.666 
27 -30 19) 12 31 40 31.319 110.485 110.523 10.862 

“27 30 | 20 18 35 00 38.044 120.659 129.574 14.795 
30 E10 25 30 18.819 83.168 83.401 6.779 
30 19 11 31 40 28.762 101.418 101.127 9.903 
30 20 12 35 00 34.937 110.806 120.178 13.837 
32 20 18 10 28 30 21.762. 85.687 89 .'792 8.376 
Sen co 20 ll 35 00 32.476 103.001 108.734 12.234 
35 20 10 | 35 00 30.071 95.372 97.115 10.574 





286 


TABLE XL.—EQUAL LENGTHS 


SELECTED SPIRALS FOR A 2° CURVE 


























A 8 nXc Ds(n+1) D’ d 
10° 1° 00’ 3X 32 2° 05’ 00” 2° 03’ 41.12 
10 1 40 4X 8d 2 08 13 2 09 61.04 
10 2 30 5 X 48 2 19 33 2 18 73.69 
10 3 30 6 X 45 2 35 34 2 33 78.81 
10 4 40 7X 44 3 01 50 2 40 10.47 
20°° 1 00 3 X 33 2 01 13 2 01 45.28 
20 1 40 4X 41 2 Ol 57 2 02 73.85 
20 2 30 5 X 48 2 05 00 2 05 99.99 
20 3 30 6 X 50 2 20 00 2 06 | 109.52 
30° 1 00 3X 34 1~ 57 39 2 Ol 46.14 
30 1 40 4X 41 2 Ol 57 2 Ol 75.16 
30 2 30 5 X 49 2 02 27 2 02 | 109.78 
30 3 30 6 X 50 2 20 00 2 02 | 115.63 
30 3 30 6 X 50 2 20 00 2 03 | 110.90 
40° 1 00 3X 35 1 54 17 2 Ol 46.90 
40 1 40 4X 42 1 59 02 2 Ol 76.96 
40 2 30 5 X 50 2 00 00 2 Ole | 017.87 

SELECTED SPIRALS FOR A 4? CURVE 
10° 14 00° 3 X 16 4° 10’ 03” 4° 07’ 20.22 
10 1 40 4x 19 4-23 13 4 16 29.12 
10 2 30 5 X 22 4 32 48 4 39 38.75 
10 3 30 6 X 23 5 04 26 5 17 41.37 
20° 1 40 4X 20 4 10 03 4 04 34.92 
20 2 30 5 X 24 4 10 03 4 09 50.72 
20 3 30 6 X 27 4 19 19 ee 63.69 
20 4 40 7X 30 4 26 44 4 3l 78.07 
20 6 00 8 X 3l 4 50 24 4 46 81.88 
20 7 30 9 X 82 5 12 36 5 16 85.40 
30° 1 40 4X 20 4 10 03 4 02 35.57 
30 2 30 5 X 25 4 00 03 4 04 57.39 
30 3 30 6 X 28 4 10 03 4 07 72.37 
30 4 40 7X 32 4 10 08 4 14 93.09 
30 6 00 8 X 35 ihe 82 4 23 | 110.31 
30 7 30 9X 37 4 30 20 4 34 | 122.20 
30 9 10 | 10X38 4 49 33 4 47 | 126.86 
40° 2 30 5 X 25 4 00 08 4 02 58.91 
40 3 30 6 X 28 4 10 08 4 04 73.75 
40 4 40 7X 382 4 10 03 4 08 94.65 
40 6 00 8 X 36 4 10 08 4 12 | 121.38 
40 7 30 9 X 39 4 16 28 4 17 | 142.86 
40 9 10 | 10x41 4 28 2l 4 26 | 154.34 
60° 2 30 5 X 25 4 00 03 4 Ql 59.68 
60 3 30 6 X 29 4 01 26 4 02 81.04 
60 4 40 7X 382 4 10 03 4 03 99.59 
60 6 00 8 X 36 4 10 03 4 05 | 125.81 
60 7 30 9X 40 4 10 03 4 08 | 154.42 
80° 2 30 5 X 25 4 CO 08 4 01 58.29 
80 3 30 6 X 29 4 01 26 4 0l 82.82 
80 4 40 7X 33 4 02 28 4 02 | 106.99 
80 6 00 8 X 37 4 03 17 4 03 | 135.61 
80 7 30 9X 41 4 03 57 4 05 | 164.79 























— 
w 
— 
oO 














TABLE XL.—EQUAL LENGTHS 


SELECTED SPIRALS FOR AN 8° CURVE 








8 nXe Ds(n+1) D’ d ze h 
2° 30 5X ll 9° CB’ OL” 9° 06’ 19.95 -6198) 051 
2 30 5 X 12 8 20 26 8 16 25.71 .9598| .051 
3 30 6X 14 8 20 26 8 34 34.86 | 1.852 gly 
4 40 7X 15 8 53 51 8 54 39.90 | 3.053 . 185 
6 00 8 X 16 9 23 07 9 24 45.52 | 4.744 eel 
2 30 5 X 12 8 20 26 8 07 26.50 .9598| .049 
3 30 6 xX 14 8 20 26 8 14 36.16 | 1.852 142 
4 40 7X 16 8 20 26 8 26 47.01 | 3.256 .260 
6 00 8x 17 8 49 55 8 36 53.13 | 5.040 820, 
7 30 9 X 18 9 16 08 8 46 60.05 | 7.452 . 287 
9 10 | 10X19 9 39 36 9 14 65.70 | 10.620 .590 
2 30 5 X 12 8 20 26 8 04 26.98 -9598| .079 
3 30 6X 14 8 20 26 8 08 36.85 | 1.852 18] 
4 40. 7* 16 8 20 26 8 14 48.25 | 3.256 293 
6 00 8 x 18 8 20 26 B22 61.385 | 5.387 . 830 
7- 30 9x 19 8 46 49 8 30 68.07 | 7.866 AT2 
9 10 | 10X 20 9 10 34 8 40 75.01 | 11.179 .629 
11 00 | 11 xX 21 9 32 03 8 54 82.13 | 15.415 . 840 
13 00 | 12 X 22 9 51 36 9714 89.81 | 20.723 | 1.024 
2 30 5X 12 8 20 26 8 02 27.30 .9598| .136 
3 30 6 X 14 8 20 26 8 03 38.22 1.852 083 
4 40 7X 16 8 20 26 8 06 49.75 | 3.256 .Ol7 
6 00 8 xX 18 8 20 26 8 10 62.87 | 5.337 639 
7 30 9 X 20 8 20 26 8 16 77.16 | 8.280 . 863 
9. 10;,{,.10 X% 22 8 20 25 8 24 93.05 | 12.297 | 1.189 
100°) ll x<'23 8 42 13 8 3l 101.08 | 16.883 | 1.523 
138 00 | 12 X 25 8 40 28 8 48 118.19 | 23.548 | 2.160 
15 10 | 13 X 26 8 58 59 9 02 127.21 | 30.817 | 2.613 
17 30 | 14x 27 9 16 07 9 22 186.45 | 39.595 | 3.157 
4 40 7X 17 7 ~50 ‘57 8 04 57.04 | 3.460 . 866 
6 00 8 X19 7 54 03 8 06 71.78 | 5.638 .408 
7 30 9X 20 8 20 26 8 08; 79.18 | 8.280 . 860 
9 10 | 10X 22 8 20 29 8 13 95.23 | 12.297 | 1.346 
li 00 | 11X 24 8 20 25 8 19 112.67 | 17.617 | 1.719 
13 00 | 12 X 26 8 20 25 8 28 180.86 | 24.490 | 2.738 
15 10 | 13 X 27 8 38 59 8 34 140.88 | 32.002 | 3.119 
17 30 | 14X 28 8 56 13 8 42 | 150.55 | 41.062 | 3.809 
SELECTED SPIRALS FOR A 16° CURVE 
4° 40’ 7X 10 13° 21’ 48” 18° 00 33.59 | 2.0385 388 
6 00 8 X 10 15 02 34 17 14 36.14 | 2.965 430 
7 30 9X 10 16 43 31 16 32 38.47 | 4.140 436 
9 10 | 10x11 16 43 31 16 48 46.40 6.148 576 
11-00. ,} 11X12 16 43 31 17 14 54.62 | 8.808 . 860 
13 00 | 12x12 18 07 48 17 22 54.14 | 11.303 | 1.093 
15 10 | 13x 13 18 O01 18 18 10 62.88 | 15.409 | 1.516 
17 30 | 14x13 19 19.14 18 12 62.85 | 19.064 | 1.552 
20 00 | 15x14 19 06 05 20 00 72.14 | 25.081 | 2.182 
7 30 9X 10 16 43 31 16 16 39.74 4.140 028 
9 10 |10X1ll 16 43 31 16 26 47.49 6.148 -680 
11 00 | 11x12 16 48 31 16 38 56.19 8. 808 948 
13 00 | 12x13 16 43 30 16 56 65.24 | 12.245 | 1.384 
15 10 | 13%14 16 43 29 17 22 74.72 | 16.594 | 1.973 
17 30 | 14x14 17 55 44 17 24 75.02 | 20.531 | 1.939 
20-00 | 15x 15 17 50 54 18 06 85.15 | 26.819 | 2.657 
22 40 | 16X15 18 58 25 18 08 85.18 | 32.276 | 2.677 
28 30 | 18x 16 19 53 20 19 42 95.84 | 48.221 | 3.748 









































288 


TABLE XLI.—GEOMETRICAL PROPOSITIONS 





The References are to Davies’ Legendre, Revised Edition. 














No REFERENCE. HYPOTHESES, CONSEQUENCES. 

HE Nes 9.0 ea ie A Ifatriangleisright | The square on the hypothe- 

angled, nuse is equal to the sum of 
the squares on the other 
two sides. 

2} 1, XI, Cor.1....|If a. triangle is | It is equiangular. 
equilateral, 

oa) ale Dag Gl A Soon ae aah 3 4 If a _ triangle is | The angles at the base are 
isosceles, equal. 

4 | L, XI, Cor. 2 If a straight line | It bisects the vertical angle. 
from the vertex And is perpendicular to the 
of an_ isosceles base. 
triangle bisects 
the base, 

5 | 1, XXV., Cor. 6.. | If one side of a tri- | The exterior angle 1s equal to 
angle is pro- the sum of the two interior 
duced, and opposite angles. 

Ge) LV. fe _X., J. ae us. If two triangles are | They are similar. And their 
mutually equian- corresponding sides are 
gular, proportional. 

Wee ST XeX VL L See If the sides of a |The sum of the exterior 
polygon are pro- angles equals four right 
duced in the angles. 
same order, 

8 | I., XXVI., Cor.1.|If a figure is a | Thesum of the interior angles 
quadrilateral, equals four right angles. 

Sie Pon KORO VEL Lae reat If a figure is a | The opposite sides are equal. ; 
parallelogram, The opposite angles are 

equal. It is bisected by its 
JEEP OO.G diagonal. And its diagonals 
bisect each other. 

10m. AIL AMILe. (0 ae If three points are | A circle may be _ passed 
not in the same through them. 
straight line, 

Ter OTL XN AL eran If two ares are in- | They are proportional to the 
tercepted on the | corresponding angles at the 
same circle, centre. 

12 | V., XIII., Cor. 2..| If two arcs are | They are proportional to their 
similar, radii. 

BT PGV eye CULL. Se coe cee If two areas are | They are proportional to the 
circles, squares on their radii. 

TAS) SELT SUV Tek iia ose If a radius is per- | It bisects the chord. And it 

j pendicular to a bisects the are subtended 
chord, by the chord. 

Loa LET SEU Sim ee cee If a straight line is ; It meets itin only one point. 
tangent to a And it is perpendicular to 
circle, the radius drawn to that 

| point. 

16 | I., XIV., Cor... | If from a point | There are but two. They are 
without a circle equal. And they make 
tangents are equal angles with the chord 

Te Xe es Sh drawn to touch 








the circle, 








joining the tangent points. 


TABLE XLI.—GEOMETRICAL PROPOSITIONS 


289 


The References are to Davies’ Legendre, Revised Edition. 








No REFERENCE. HYPOTHESES. CONSEQUENCES. 

Vet EELS XS 2855 oot: If two lines are | They intercept equal arcs of 
parallel chords a circle. 
or a tangent and 
parallel chord, 

ee ALL; VIL... .. | If an angle atthe | The angle at the circumfer- 


circumference of 
a circle is sub- 
tended by the 
same are as an 
angle at the cen- 
tre, 

19 | II1., XVII.,Cor.2 | If an angle is in- 
scribed in a semi- 
circle, 

205) LAE oO. 6 Ue eee Ii >tan “angie —is 
formed by a tan- 
gent and chord, 
21} IV.,XXVII.,Cor. | If two chords in- 
tersect each oth- 

er in a circle, 


And if one chord is 
a diameter, and 
the other  per- 
pendicular to it, 


22 | IV., XXIIL.,Cor.2 


PalelVn SLX Cor. If tivo secants 
meet without the 
circle, 

PAIL Vc XGX. OD. eee os (IE, a “Secant and 
tangent meet, 


25 If a straight line 
from the vertex 
of a triangle bi- 
sects its base, 


eset re cee 


re 
or 
bi 
La 
< 





26e/01 Ve. If a perpendicular 
be drawn from 
the vertex of a 
‘s triangle to the 


base, 








ence is equal to half the 
angle at the centre. 


It is a right angle. 


It is measured by one half of 
the intercepted arc. 


The rectangle of the seg- 
ments of the one, equals the 
rectangle of the segments 
of the other. 


The rectangle of the seg- 
ments of the diameter is 
equal to the square on half 
the other chord. And the 
half chord is a mean pro- 
portional between the seg- 
ments of the diameter. 


The rectangles of each secant 
and its external segment 
are equal. 


The rectangle of the secant 
and its external segment is 
equal to the square on the 
tangent. And the tangent 
is a mean proportional be- 
tween the secant and its 
external segment. 


The sum of the squares on 
the two sides is equal to 
twice the square of half the 
base increased by twice the 
square of the bisecting line. 


The square of a side opposite 
an acute angle is equal to 
the sum of the squares of 
the other side and the base, 
diminished by twice the 
rectangle of the base and 
the distance from the ver- 
tex of the acute angle to 
the foot of the perpendicu- 
lar. 


EEE et 


290 TABLE XLII.—TRIGONOMETRIC FORMULAS 


TRIGONOMETRIC FUNCTIONS. 


Let A= angle BAC = are BF, and let the radius AF = AB= AH = 1. 
PA Ee 
We then have 


sin 4A = BO 
cos 4 = AC 
tan A =D 
cot A = HG 
sec A = AUD) 
cosec 4 = AG 


versin 4 =<CF= BH 
covers A = BK= HL 
exsee A 
coexsec A = BG 
chord A = BF 
chord 2 A = BI= 2BC 


II 
ee 
o 





In the right-angled triangle ABC. 
Let AB =c, AC = b, and BC= a. 
We then have: 





1. sin A = < =cosB HS a =e sin A = b6tame 
b ‘ 
2. cos A Saber sen: sin B 12. +b:= ¢ cos 44a cots 
ates CaN 7) a See 
3. tan A = ame cot B Chea oe sin cde ae A 
b 
4.) \e0tcA nat Shrines tan B 14. a =cecosB=bcotB 
c 
5. sec A be pti ss ecosec B 15. 0 = csin B= ee tanve 
c a b 
6, cosecA = chine sec B 167 eee bse nee 
” A £3: G@sab= vers B 17 ee (ae a7 
, vers i eae rs = covers ‘ = V (c+ b) (ec — dD) 
eed | : Lge: ie es 
8. exsec 4d = pb = coexsec B 13h (Oana (c+ a) (e— a) 
9, covers 4 = ~~“ — versin B BA ll ce tiee Voor wily 0) 
. “4 C eh a2?-+b 
10, coexsec A = © — = exsec B 20; C= 90° =4 - B 





21. area = oe 





TABLE XLIU.—TRIGONOMETRIC FORMULAS 


— 





SOLUTION OF OBLIQUE TRIANGLES. 
B 












































a? sin B. sin C 
3 | A, B,C,a | area K= dan A 


291 


eet 


GIVEN. SOUGHT, FORMULAS. 
; ; ieee, " Sas : 
22 | A, Ba CL b,.¢6- "C= 180° — (Aaa; b =o sin B, 
a 
C= Sac sin (4 + B) 
£38 | A, a, 0 B,C,c¢ |sn B= = = ou ay. C= 180° —-(4+ B), 
= eA sin C, 
24°) CC, a, 6 (4+ B)|/4(44+ B)=90°-YwYO 
2 146 (A — B) fan $e caylee B= ap tan 34 (4 + B) 
26 A,B A='’%(4+ B)+1%(A4 — B), 
B=4%(A+ B)-W%(A-B) 
- cosl4(A+B) Sin 1e(A-+ B) 
he E OF Oa costa apy ~~) singed By 
» (OF area kK='WabsinC. 
p (s~b) (s—c) 
29 | a,b,¢ A |lets=34(a+b+e);snkda=4/S— 7 S—O 
s(s (s—b\(s—e 
30 cosl4A =4/ 55 tanlg A= so ac 
; 24s (s — a) (s—b)(s — ©). 
31 sin A.= oo 
is Weg e UE ©) 
be 
32 area =| K = Vs(s — a) (s — b) (s—e) 





TABLE XLII.—TRIGONOMETRIC FORMULAS 


GENERAL FORMULAS. 


a 























34 | sn A = —- ety 7¥1—cos?A = tan AcosA 
cosec A 
35 | sn 4 = 2sinlgAcoswWA = versAcothA 
36 |snA = Vigvers2A = Y16(1—cos2 4A) 
37 | cos A = Berta Vv ‘ -—sin? A = cot Asin A 
im ‘ee See Aes : 
88 | cos 4 = 1—versd4 = 2ecos?l44A—1 = 1—2sin? lg A 
39 | cos A = cos?hA—sin?, A = V¥4%+%cos2A 
i sin A — 
Bye oe a a gs cac2 4 — 
40 | tan Ao = a G65 A VW sec? A 1 
Sah .. 7 1— cos? A sin 2.4 
cA ce aa 4/ we AE ee cos A” 1 Gos Ola 
1—cos2A vers 2A 
42 | tan A = ie SOD we errr Ss exsec A cot 14 A 
1 cos A a 
ay, Seer eS 2 Sie 
43) | cot 4 = tana ies vy cosec? A —]1 
; sin 2.4 sin 2A itcos2A 
of ce et coe ad vers 2.4 sin 2.4 
48 | cot 4 = Snes: 
exsec A 
46 | vers A = 1—cosA = sinAtanljA = 2sin?kg A 
47 | vers 4 = exsec A cos A 
48 | exsec A = secA—1 = tan a taniA = sea 
- be // oem 1S | 
a9; | giasde a 1—cos A est A 
2 2 
50 | sin2A = 2sin 4 cos A 
51 | cosgA = {/ 
na| cos 84 = 2cos?A—1 = cos? A—sin? 4 = 1—2sin° A 





TABLE XLII.—TRIGONOMETRIC FORMULAS 293 





GENERAL FORMULAS. 














53. tang d= ey = cosec A — cot A = bets = 205 2 4 oe 
b4. tan 2.4 9 Sales a 

BB. cot. WA = TOA = “sind ea 

56. cot 2.4 = or 4+ 

bY. vers}4.4 = ye Ners Adi 1—cos 4 


1+ ¥i— — yg vers A 2+ V¥2(1 + cos A) 


58. vers 2 A =2 sin? A =2 sin A cos A tan A 


1—cos A 


(1+cos A) 4V XL: ‘cbs A) 





59. exsee 4 A= 


2 tan? A 
60. exsec 2 A = 
1 — tan? 4 


61. sin (4 + B) = sin A.cos B + sin B.cos A 

62. cos(A + B)= cos A.cos B F sin A. sin B 

63. sin A + sin B = 2sin14(A-+ B)cosl4(A — B) 
64. sin A — sin B = 2cos%(A-+ B)sin4(4 — B) 
65. cos A + cos B = 2cos 46(4 + B)cos 16(A — B) 


66. cos B— cos A = 2sin144(4-+ B)sin 144(A — B) 


67. sin? A — sin? B = cos? B — cos? A = sin(A + B)sin (A — B) 
68. cos? _A — sin? B = cos (A+ B) cos (A — B) 


sin (A + B) 


69. tan 4 + tan B= oo cos B 


sin (A — B) 


70. tan A ah B= on 4 cos B 





294 TABLE XLIII.—FORMULAS 
pearance 


SERIAL} TEXT 




















nel tong GIVEN. FORMULAS. 
SIMPLE CurvE ForRMULAS 
OU 
1 > v ~ gin 4D 
50 
2 17 R sin 4D = = 
ae ads aD dia pe = 
; vi D 
DL 
4 19 Dr 500 
A 
5 20 A, BL D= 100 = 
L 
6 21 Ran T=R tan YA 
7 22 a C=28# sin WA 
8 24 a M = R vers %A 
9 24 ve E = R exsec 4A 
10 25 TaN R=T cot 4A 
ll 26 Ne E=T tan YA 
ene ese a C= 2T cos ’A 
13 leer et a M =T cot WYA-vers “4A 
E 
E — 
2 gl re exsec 4A 
15 28 ce T = EF cot WA 
16 29 R,d c= 2R sin 4d 
17 32 d, D Ce 100 £ (approx.) 
_ (100)? 
18 37 R A= af 
C2 
19 38 R, c 4 = OR 
e2t 
20 39 R, c th (100)2 
Bp 
21 55 Bp, A AA’— BB’ = — 
sin A 
as ; Bp 
vers A 
/ 
23 | 59 BREA OR Ra oe 
exsec A 








SASS SSE Rae eae sag ETE I oem ce 


CURVES AND EARTHWORK 


295 





SERIAL 
NO, 


24 








GIVEN. | FORMULAS, 











SimepLE Curve FormMvuuas 





60 Rh, Re’, 3A AA’=(R-—R’) tan A 


sin A 
exsec 4A 


ME ae ce eo M = E cos ¥A 


4 Li Cula 
~ 2sin 4A 


ae Vewes Se M=4C tan YA 


Ly LA = 
~ 2 cos YA 


Pia E, A C=2E 








Beis ets ae T 
exsec 4A 
sin 4A 


M 
ep vers 4A 


Fie! ivcs 4 E='C 


ate Sede pes C=2M cot YA 





“s ny tan 4A 
BAe T= Moon YA 


“ ew 
Ma. Teteen VAN 


Be ae RL tan YA=z 








296 TABLE XLIII.—FORMULAS 





SERIAL) TEXT GIVEN: FORMULAS. 















































NO. NO. 
SIMPLE CuRVE FoRMULAS 
44 TC bos Bea ee 
Ri eee : oT 
2T-—C 
ag 1 me 
Aly alle Beaters: tan 4A f/ a G 
AG Ga llaherste ee 1M 1D tan YA = E 
L 
“ _ T2— K2 
CM foes Rep seme . cos 4A = TB? 
Ey eae Cc, M on ia ri 
C2— 4M2 
66 y% as 
AQ Ral ere ae cos 4A C24 4M 
M 
OO ll dee sai M, E cos 4A = a 
E— Mo M 
66 yy 
Suen ese se tan 4A eee 
Rv = 20 
D2 TA ticee te # Were ai 
R2 
te M = k —- —— 
DS Pal cae. he ae a 
SS is Ei Se oi E=VT+R2—R : 
CR 
DD aul teeponks Tee AO T= A = 
24/ (2+5) (#-§) 
Boca ae A M=R-—V(R+4C)(R— %O) 
R2 
Bi ee » BS = 
(R+4C)(R — 4C) 
| fe RV M(2R— M) 
DS Se ee ws. Rk, M = R—M 
aD eens es C= 2VM(2R— M) 
RM 
60 eee e ee | E= fae Yj 
AS eee R, E T =VEQR+E) 
VR(OR LE\ 
COE hse zr cu 2RVEQR+ E) 
R+E£ 


: epee irae. Sere 


CURVES AND EARTHWORK 297 









































SERIAL| TEXT 
NO. NO. GIVEN. FORMULAS. 
SimpLe Curve ForMULAS 
RE 
63 | FRR tc. R, E M= PLE] 
reg ae ee a Be CO ee 
V(2T+ C)(2T—C) 
OTC 
sé amt 
ia eee eae M Ke4/ Faro 
De 2T— CG 
eke ee E= 7 Vee 
_ (TL E\(T- EB) 
(ee ae: fd 2 Ra 
Pe _ 27(T?— E?) 
68 Si Ce ae 
oe = E(T?— E?) 
69 ae es 
. M?*+(KC)? 
BO Pu. Pus Cc, M R= 
. _ O(02+ 4M?) 
TE We suived Te sete ais 
~ _ 4, C?4+4M? 
0 dhs Pi, B= Mine agi 
EM 
pee hee ee M, E a eta 
we re E+ M 
74 4}. tow. ; | Da 8 er a 
Pp if E+M 
FEE | hee C= om 4/ a 7 
bd 
eT and £7, M rR LRT? —VMT?=0 
(ag Cea iy ee 3 * E34 E2M — ET?+ MT?= 0 
be ae eae C34 2702+ 4M°C— 8M?T= 0 
42 — C2 G?. CE 
Woe Mie CAE Beg ht i ea Ba 
SN it. Sas ais ss 273 — T2C — 2TE2?— CE?=0 
C2 C2E 


SiH. fate « ; MiG Mn+ MH ~=—=0 





298 TABLE XLIII.—FORMULAS 
































SPRL oe GIVEN, FORMULAS. 
COMPOUND AND REVERSED CuRVES 
82 | 100 | Si, &, A | cot y= 2S cot rsa 
S2— Si 
838 | 101 Ai, As A= Ai+ Az 
Si, So, Ai, = Y4a(S2 re S1) (cot vay TGOw Y2A1) 
84 102 
{| Ag ¥ = 14(S2— S1) (cot 4y + cot 14A2) 
85 | 103 | St, S2, Ai, Ae | Ra— Ri = }4(82— Si) (cot 4A + cot 14A1) 
Ri 
hi, Ra, 3% ae oct aha 
86 | 104 ae - 
1, S2, 7 SA ce oe 
cot Ae VS — SD cot ky 
YWAB | sin 4y 
1 = = 2 = 
87 105 AB, A, 72(S2 — 81) sin 144A 
Ro2—-Ri 
1 = cS 3y gee oaths gee 
BO | 106 | | Peay Bay At Saal aS lle Se cca 
Ri 
cot by = 4 (Sp =n Sp) aa cot YA 
Ri, Re, S81, F 
Sy af 10 7. 
2, Ai, Zz 1 1 
cot By = (8 — Si) 5) — cot YA2 
¥o(Se— Si) cot Yy 
1 = 
90 | 108 | St, S2, A, y | 24(S2+ S1) cot 4A. 
16 (S2— S1) sin 4A 
91 | 109 | Si, S:, A, 7 | MAB= sin My 
1 
92 | 206 Dp, 7 MA 2dr ges ae 
~ * en 
93 207 DP, Ar vers A, 
94 | 208 rehTie oe 
Pp, Ar vers A, 
95 209 P, Ti, 72 vers A, rita 
96 214 t, Ai, Ae : 




















= LS 
tan 4%Ai+ tan Was 


SERIAL 





TEXT 





CURVES AND EARTHWORK 


GIVEN. | 





299 











NO. Se FORMULAS. 
TURNOUTS 
97 220 F n= % cot 4F 
98 223 |l,¢,h,f,t,F,n| Lg=l+[g@—h—f sin F)] 
cot K(F + S8)+/f cos F+ nt 
n We g—hA—fsin F 
peal peest te Gael 2S tt 40= 5 in (RF — S) sin 4(F 4-8) 
106 232 le Megs o=h— (r+ Y%g) vers S 
101 240 Goan L, = 2gn 
102 241 g, n r= 2gn?= Ln 
ce Ps Oe 
103 244 ja coy al a sie cot F 
104 246 ey Ni. LS tee oa=g+k sin F—(r+ 4g) vers F 
7 Pg Pao sin KF 
105 250 p, 9, k, F ri — log oF 
106 271 (fe, Py A o=g—(r+%g) vers F 
SPIRALS 
SAT 
107 | 284 | 2, 2, uy, y | tana=—— = 
108 285 ‘ns 1=a—s 
109 | 286 x, y, 4 SL=— = 
cost sini 
110 | 287}. ews LE=——, SE=y~xcots 
sin s 
Uy 290 ae Tee p=xa—R’ vers s 
112 291 pleas q=y—R’ sins 
113 | 292 eR a ve ry eo tA A 
y; : ’ ’ a y = cos Vea 
114 294 Leer me T, =R’ tan 4A+ ne (approx.) 
; . _ «+R’(cos s— cos 4A) 
115 295. |) %, hi’, 8, A} Hy ccs | 
116 299 |R, R’, z, s, A} heos4A=(R—R’) vers 4A =(x —R’ vers s) 
117 300 If 1s UM IP p= (R— R’) vers 4A—h cos WA 
118 301 ly, R,R’,s,h, A) d@d=y—R’ sins— (R—R’-+h) sin WA 
i eee, Vers FF LD 
119 306 x, R, 8, A —h= cos 6A ~~ eos WA 
120 307 |: a, h, p, a d=q+thsin 4YA=q+>p tan 4A 
, Pe Sa OE OS nyse fined a Yeas 1h 
121 308 Died 9) 2 oa sin A tanA val sinA tandA 
122 309 |¢,¢53 AC, BC’) d=q+AC d’=q'+ BC’ 
a p 
123 315 p, A h= dis if A 
124 316 T,q, p, S| T,=T+at+ptan YA 















































300 TABLE XLIII.—FORMULAS 




















SERIAL| TEXT 
wo, Sey. GIVEN. | : FORMULAS. 
EARTHWORK 
125 331 b, 8,"d 2£=b4+sd 
x= b+ sh 
126 |. 332% feds mk ee: 
= 4b+4+ sk 
127 336 b, s, d A = bd4sd?2 





128 | 337 | b, mn, d | A=%b+m4n)a 
129 | 340 |b, 4, b, m,n} A= Md(m+n)+4b(h+h) 





b 62 
34 b, ’ d, ’ =f L Be D1 sae Nae 
130 1 |b,s,d,mn| A=¥% (a+ =) (m+n) — © 





131 342 pled 6 Nal Bek ar 














27 
132 | 843 | 4, A’, say dM 
M | S=5yy7 (At4M+ 4) 
133 344 SE, SP C= SE—Sp 
8451348 |e pce pte Ee es 
Os ascay Or GiB) 
135 | 348 |d, a’, D, D’ l , 
CATR He oe 
136 | 352 WK gat (Bet) 
37 3 
137 353 AK gua (Geet eth) 
27 4 
138 | 354 A,s S a= 4 ith 2st 883+ 4e4 
4X 27 
139 | 355 bag caret, - Ore + ad) 
50 
357 b 
140 mG = 54 0% 
b;\d; "mM, ’ 
141 | 358 Sis Su= 2 d(m-+n) + 22 7 bh +4) 


50 b 50 62 
359 b, 9 Uy, ’ Uy 3 aa aa a ae 
142 8s, d, m, n SE ad (a+) (m+ n) 5d Qs 


143 | 360 |d, a’, D, D’| c= rd d’)(D — D’) 


_ (A A’) 
12 X 278 


a a ere ence ee ee ee ee 


144 365 li Ay AGS 





CURVES AND EARTHWORK 301 























SERIAL] TEXT 
a a0. GIVEN. | FORMULAS. 
EARTHWORK 
ee fol Aaa 
145 | 366 Tovdl A 2 ie, 
146 368 ° CATA M=C-tan\Ya 
147 369 D m= 16100 tan 4D 
; 
148 | 370 R, ¢ M=R = oe (5) 
3 R ch 
149 37 ae m = 3k (approx.) 
150 375 n IDE a (approx.) 
151 376 n Di a (approx.) 
, raigiS g’ 
152 377 9, 9', 4 Le Se 
r OY ieee eid 
153 378 9,9," a= on 


154 | 381 | 9, V, R | e=.0sess 2 





302 





Title. 


Ratio of circumference to diameter 


ee wee 


Recrprocalfolsamenwwi ee ean: 


Degrees in arc of length equal to radius. 


Minutes ‘*‘ oe AS ae: 
Seconds ‘ = $3 ey eho 
Length of 1° arc, radius ‘unity....3: 5. .° 
Length of 1’ are, i Tks teen : “ A 
Hengeth ofl sarc as “SOUR eos 


Radius by which 1 foot of arc = 1 degree. 


Radius ‘‘ Sh, A EL Thinate: 
Radius ‘‘ ‘“ xho ‘‘ ‘* =10seconds 


Factors for dividing a line into extreme 
and Mean TAtiOns.... A0s0 sone kn aes 
Base of hyperbolic logarithms.......... 
Modulus of common pxaten of logs =log e 
Reciprocal of same =hyp. log. 10....... 
Length of seconds pendulum at New 


York in inches (at 125 feet above sea 


Length of seconds pendulum at New 
York in:téet sats fn oe ve cu en et ae 


Acceleration due to gravity at New York 


Square Toot olsamen.s 0: oe eee aie 


Wards in alsin eters ke eo ee 


eG mmc lame 


een eeeeecsrecercec eee eeceeese 


Inches in 1 


eeeoe cece reveeoceeweence 





Symbol. 





10800’ 


TT 





-648000’” 





SiS 


9 
or 





Number. 


3.1415927 
0.3183099 


57. 295780 
3437 . 7468 
206264. 81 
- 01745329 


. 00029089 


| 
-000004848 


57. 295780 
343 . 77468 
206. 26481 
0.6180340 
0.3819660 
2.7182818 
0. 4842945 
2.3025851 


39. 10299 


3.25858 
32.1609 


5.67106 
1.093611 
3. 280834 
39.37 
0.304806 
0.914418 


1609. 344 











TABLE XLIV.—USEFUL NUMBERS AND FORMULAS 


Loga- 
rithm. 
0.4971499 
9. 5028501 


1. 7581226 
3.5362739 
5. 3144251 
8. 2418774 
6. 4637261 


4. 6855749 
1. 7581226 
2.53862739 
2.3144251 
9.7910124 
9. 5820248 
0. 4342945 
9. 6377843 
0. 3622157 


1.5922100 


0.5130244 
1.5073282 


0. 7536642 
0.0388629 
0.5159843 
1.5951654 
9. 4840235 
9.9611448 
3. 2066489 


Ne yc ee ee. 
































TABLE XLIV.—USEFUL NUMBERS AND FORMULAS — 303 
Title. symbol. | Number, | j/084 
Cubic inches in 1 U.S. gallon .............. 231, 2.3636120 
A [ele erm periale gallon ics scr 277.274 | 2.4429092 
ss 2 SES Fh WO otek SUITE! Dee aie ci Aeaoe 2150.42 | 3.3825233 
@ubic feeb in! # Uso. eallon ss nasa cic c= eae 0.133681, | 9.1260683: 
4 eee olin periale y aulOMs. entails la 0.160459 | 9.2053655 
ie oma 12 Gras, OUISHOL cat alee elec cee 1.244456 | 0.0949796 
Weight of 1 cub. foot of water, barom. 30 in. 
ther. 39°.83 Fah.; pounds.. 62.379 | 1.7950384 
en Oee ay Sc 62.3821 | 1.7946349 
Weight in artes 1 cubic inch, at 62° Fah.. 252.458 | 2.4021892 
No. of grains in 1 pound avoir.............-. 7000. 8 ..8450980 
Ss eT OTnCG Say hoop neds fa 437.5 | 2.6409781 
1 Se veel ge 
y = radius of circular arc; in W 
l = length of arc; (is < — 
a° = degrees in same arc, | W 
t= aor. 7808 


/ 
Radius by which the length of chord c in feet = a in minutes; 


tea’ 
10 sin 4a’ 








Hyp. log a = com. log x X a OL 


com. log (hyp. log x) = com. log (com. log 2) + 0.3622157 
Com. log « = M X hyp. log x; or 


com. log (com. log x) = 9.6377848 + com. log (hyp. log x) 


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EXPLANATION OF TABLES 


TaBLE I.—Contains the radius, and its logarithm, for 
every degree of curve to single minutes up to 10°, and thence 
by larger intervals up to 50°. With the radius is given also 
the perpendicular offset, ¢, from the tangent to a point on 
the curve at the end of the first 100-foot chord from the 
tangent-point, and the middle ordinate, m, of a 100-foot 
chord.. See eqs. (16, 34, 37, 40 and 369). 


TaBuE I].—Contains the corrections to be added to the tan- 
gents and externals of any railroad curve, as obtained by 
reference to Table II], according to the degree of the given 
curve (found at head of columns), and its central angle, 
(found in the first column). If the given degree of curve, 
or central angle, does not appear in the table, the exact 
value of the correction may be easily obtained by interpo- 
lation. 


Taste I1].—Contains the exact values of the tangents, 
T, and externals, E, to a 1° curve, for every 10’ of central 
angle, from 1° to 120° 50’. Approximate values of the 
tangent and external to any other degree of curve may be 
had by simply dividing the tabular values opposite the given 
central angle by the given degree of curve, expressed in 
degrees. These approximations may be made exact by 
adding the proper corrections taken from Table II. See 
eqs. (21) and (24). 

TasBLE 1[V.—Contains the value of Long Chords of from 
2 to 12 stations, for every 10’ of degree of curve from 0° 
to 15°, and of a less number of stations for degrees of curve 
between 15° and 30°. As the chord of 1 station is always 
100 feet, the column of the first station gives instead the 
length of arc subtended by the chord of 100 feet. See 
§§ 121, 122, 423, 124, 125 of the text. 

Taste V.—Contains the chords of a series of angles 
varying by half degrees up to 30° fer radii varying by 100 

305 


306 FIELD ENGINEERING 


feet up to 1000 feet. It shows, therefore, the linear opening 
between the extremities of two equal lines at any given num- 
ber of hundred feet from their intersection, when the angle 
does not exceed 30°. For any distance exceeding 1000 we 
have only to add to the value found in that column the value 
found in the column headed by the excess of distance over 
1000 feet. Conversely, the table gives the angular deflec- 
tion required between two equal lines, in order that at a 
given distance from the point of intersection they may be 
separated a given amount. 


Tasie VI.—Contains the values of Middle-Ordinates to 
long chords of from 2 to 12 stations, for every 10’ of degree 
of curve from 0° to 10°, and of from 2 to 6 stations for every 
curve from 10° to 20°, at 10-minute intervals. The table 
may be used, not only to fix the middle point of an are, 
but also, in conjunction with the table of long chords, to 
locate intermediate stations. See §§121, 122, 123, 124, 
125 of the text. 

TaBLE VII.—1. Contains values of the ratio u = = 
according to the notation of § 147 for finding the angle 7 
(Fig. 34) between the radius PO of the curve at any point P, 
and the tangent PK to the valvoid are PX by the simple 
formula eq. (80) 7 = uA. The table embraces lengths of 
curve from 300 to 2000 feet, and central angles from 10° 
1G5120 2. 


When an = 60°, u = 1 and for hasty approximation 


this value of u may be assumed in any case without con- 
sulting the table. 


2. Contains values of the ratio » = & 


L 
radius of the valvoid are at the point P (Fig. 85) in terms 
of the length of curve L = AP by the simple formula, eq. 
(82), r = vL. 

3. Contains values of the length 1, of a valvoid are limited 
by two curves of equal length laid out from the same tan- 
gent and same P.C., but whose central angles differ by 1°. 
The length L of each curve is given in the first column, and 


the half sum of their central angles (Ate 


for finding the 


9 ) is given at 


the head of the other columns. 


EXPLANATION OF TABLES 307 


When the central angles of two curves of equal length 
differ by x degrees the length / of the valvcid arc joining 
their extremities is expressed by the simple formula, Fig. 36, 
eq. (86) 

bag? Pe (A’ ih A 


in which J; is taken from the column headed by 5a ee 


and opposite the given value of L; or J, is found by inter- 
polation if necessary. See § 150 and example. 

TaBLeE VIII.—Contains the middle-ordinates of chords 
varying in length from 16 to 60 feet, and for degrees of curve 
varying from 1° to 50°. The use of the table is obvious. 
' See § 315. 


Tapte IX.—Gives the proper difference in elevation of 
rails on curves of various degrees from 1° to 50° for veloci- 
ties varying from 10 to 60 miles per hour. See § 328. 


TaBLE X.—Contains the values of inches and fractions 
expressed in decimals of a foot for every 32d of an inch 
up to 1 foot. 


Taste XI.—Contains the radius and its logarithm for 
the metric curve based on a chord of 20 meters, the de- 
flection angle ranging from 0° 10’ to 10°. It gives also the 
middle ordinate and tangent offset in meters for one chord, 
and the degree of curve of the identical curve if laid out 
in chords of 100 feet by the American method. Numerical 
values in this table can be derived from Table I by first 
doubling the deflection angle to get D and then multiplying 
the values in that table opposite D by 0.20. 


TaBLE XII.—Contains the acreage for right of way 
100 feet wide given to tenths of a station or for 10 feet. 
The values for shorter lengths can be found by interpolating; 
those for different widths by proportion. 


TaBLE XIJJ.—Contains the vertical heights through which 
a train moving at certain velocities would be raised by its 
own momentum. The table is based on the formula H = 
0.0355V2 which includes the effect of the rotative energy of 
the wheels. 

Taste XIV.—Gives the rise of grades in feet per mile 
and their angle of inclination corresponding to a rise per 
station (100 feet) varying from 0,01 foot to 10 feet. 


308 FIELD ENGINEERING 


TaBLE XV.—Contains values of the formula (log h — 1) 
60384.3 in which h = reading of the barometer in inches. 
The inches and tenths of the readings are in the left-hand 
column, while the hundredths are found at the top of the 
other columns. The difference of any two values correspond- 
ing to two readings taken simultaneously at any two sta- 
tions is the difference in elevation in feet of those stations. 
But the difference in height so found is subject to a correc- 
tion for temperature given in the next table. See § 10. 


TasBLE XVI.—Contains coefficients of correction for 
atmospheric temperature, by which the approximate heights 
obtained by Table XV are to be multiphed for a correction 
of these heights, which correction is to be added or sub- 
tracted according as the coefficient given in the table is 
marked + or —. See § 11. 


TaBLeE XV1I.—Contains corrections in feet, required by 
the curvature of the earth and the refraction of the atmos- 
phere, to be applied to the elevation of a distant object 
as obtained by a level or transit observation for distances 
ranging from 300 feet to 10 miles. See § 247. 


TaBLe X VIII.—Contains information for finding the differ- 
ences in elevation and the horizontal distances in stadia 
work. The table is based on the formulas: the difference 
in elevation = $ks sin 2a + (f +c) sin a; and the hori- 
zontal distance = ks? cos a + (f + ¢) cos a. 

The quantities in the table under “ Diff. Elev.”’ are values 
of 4ks sin 2a where k = 100, s = 1 and a = the ‘particular 
vertical angle. With any other values of k and s the product 
is found by use of the slide rule. The quantities at the foot — 
of the column are values of (f + c) sin a for three values of 
f +c (called ¢ in the table). The products for other values 
of (f + c) are found by interpolating. 

Under ‘“ Correction to Horizontal Distances ”’ are given 
100 — ks cos? a for different values of a, and for k = 100 
and s = 1. Proper multiplication is made for other values 
of these quantities. The value (f +c) cos a is found at the 
foot of the page for three values of (f + c). 

Example.—Angle 2° 46’; rod reading 2.64; f +c = 1.1; 
and k = 100. 

Difference in elevation = 4.82 X 2.64 + .04 = 12.76. 


*  BXPLANATION OF TABLES 309 


Correction to horizontal distance for 100 feet = .23 and 
for a rod reading of 2.46 it is 2.46 X .23 or .6. Correction 
for (f +c) cos ais 1.1. Therefore the horizontal distance 
246 — 0.6 + 1.1 = 246.5. 


TaBLE XIX.—Contains the value of the sun’s refrac- 
tion in terms of the latitude of the place and the hour of the 
observation. The hour angle indicates the distance of the 
sun from the meridian. Three hours indicates either 9 A.M. 
or 3 p.m. These times are strictly apparent solar time which 
is governed by the motion of the real sun. For all practical 
purposes, however, the time may be considered standard 
time which is kept by the observer’s watch. 

Complete directions are given in the Nautical Almanac 
for the computation of declination, and the refraction values 
are applied to the declinations thus found. North declina- 
tions are + and those south are —. The refraction cor- 
rection is to be added to the former and subtracted from the 
latter. 

This table is used in laying off the proper declination 
for ap observation with the solar attachment. 


TABLE XX.—Gives the lengths of circular ‘arcs to a radius 
=f, 

To find the length of any are expressed in degrees, minutes, 
and seconds, take from the table the lengths of the given 
number of degrees, minutes, and seconds, respectively, 
and multiply their sum by the length of the radius. The 
product is the length of are required. 

Tasie XXI.—Contains the values of minutes and seconds 
expressed in decimals of a degree, for every 10” of arc, and 
also for quarter minutes up to 1°. 

TaBLE XXIJ.—Contains the measurements necessary to 
lay down a turnout with frogs of given numbers or angles 
for both a standard and a 3-foot gauge. The distance BF 
is measured on the rail of the given track from the heel of 
the switch to the point of the frog, while af is the chord 
of the center line of the turnout between the same points. 
The radius r applies to the center line of the turnout. The 
distance aF’’ is measured on the center line of the straight 
track from the heel of the switch to the point of the middle 
frog. The length of switch AD should conform to the 


310 FIELD ENGINEERING 


tabular values unless the throw is to be different from that 
assumed in the table. See § 195, 196 and 204 of the text. 


Tastes XXIIA and XXIIB.—Contain the recommended 
standards of the American Railway Engineering Asso- 
ciation for frogs and split switches. Offsets are given for 
locating the lead curves. 


TaBLE XXIII.—Contains the squares, cubes, square 
roots, cube roots, and reciprocals of numbers from 1 to 1054. 
Its use may be greatly extended by observing that if any 
number is multiplied by n its square is multiplied by n?, 


its cube by n’, and its reciprocal by * 


Taste XXIV.—The logarithm of a number consists of 
two parts, a whole number called the characteristic, and a 
decimal called the mantissa. All numbers which consist 
of the same figures standing in the same order have the same 
mantissa, regardless of the position of the decimal point 
in the number, or of the number of ciphers which precede 
or follow the significant figures of the number. The value 
of the characteristic depends entirely on the position of the 
decimal point in the number. It is always one less than the 
number of figures in the number to the left of the decimal 
point. The value is therefore diminished by one every 
time the decimal point of the number is removed one place 
to the left, and vice versa. Thus 


Number. Logarithm. 
13840. 4.141136 
1384.0 3.141186 
138.40 2.141136 
13.84 eet 14 sg 
1.384 0.141136 
. 1384 —1.1411386 
.01384 —2.141136 
.001384 —3.141136 

etc. ete. 


The mantissa is always positive even when the character- 
istic is negative. We may avoid the use of a_ negative 
characteristic by arbitrarily adding 10, which may be 
neglected at the close of the calculation. By this rule we 
have 


EXPLANATION OF TABLES td. 


Number. Logarithm. 

1.384 0.141136 

. 1384 9.141136 

.01384 8.141136 

001384 7.141136 
etc. etc. 


No confusion need arise from this method in finding a num- 
ber from its logarithm; for although the logarithm 6.141136 
represents either the number 1,384,000, or the decimal 
.0001384, yet these are so diverse in their values that we 
can never be uncertain in a given problem which to adopt. 

The table XXIV contains the mantissas of logarithms, 
carried to six places of decimals, for numbers between 1 
and 9999, inclusive. The first three figures of a number 
are given in the first column, the fourth at the top of the other 
columns. The first two figures of the mantissa are given 
only in the second column, but these are understood to apply 
to the remaining four figures in either column following, 
which are comprised between the same horizontal lines 
with the two. 

If a number (after cutting off the ciphers at either end) 
consists of not more than four figures, the mantissa may be 
taken direct from the table; but by interpolation the log- 
arithm of a number having six figures may be obtained. 
The last column contains the average difference of con- 
secutive logarithms on the same line, but for a given case the 
difference needs to be verified by actual subtraction, at least 
so far as the last figure is concerned. The lower part of the 
page contains a complete list of differences, with their mul- 
tiples divided by 10. 

To find the logarithm of a number having six 
figures. Take out the mantissa for the four superior 
places directly from the table, and find the difference between 
this mantissa and the next greater in the table. Add to the 
mantissa taken out the quantity found in the table of pro- 
portional parts, opposite the difference, and in the column 
headed by the fifth figure of the number; also add one tenth 
the quantity in the column headed by the sixth figure. 
The sum is the mantissa required, to which must be pre- 
fixed a decimal point and the proper characteristic. 

Example.—Find the log of 23.4275. 


ei2 FIELD ENGINEERING 


For 2342 mantissa is 369587 
For diff. 185 col. 7 129.5 
For diff. 185 col. 5 Ot 





Ans. For 23.4275 log is 1.369726 


The decimals of the corrections are added together to 
determine the nearest value of the sixth figure of the mantissa. 

To find the number corresponding to a given 
logarithm. If the given mantissa is not in the table, find 
the one next less, and take out the four figures corresponding 
to it; divide the difference between the two mantissas by 
the tabular difference in that part of the table, and annex 
the figures of the quotient to the four figures already taken 
out. Finally, place the decimal point according to the rule 
for characteristics, prefixing or annexing ciphers if neces- 
sary. The division required is facilitated by the table of 
proportional parts, which furnishes by inspection the figures 
of the quotient. 

Example.—Find the number of which the logarithm is 





8.263927 8.263927 
First 4 figures 1836 from 263873 
Diff. 54.0 : 
Tabular diff. = 236 °°. 5th fig. = 2 2 
6.80 
6th fig. = 3 7.08 


Ans. No. = .0183623 or 183,623,000. 


The number derived from a six-place logarithm is not 
reliable beyond the sixth figure. 

At the end of the Table XXIV is a small table of logarithms 
of numbers from 1 to 100, with the characteristic prefixed, 
for easy reference when the given number does not exceed 
two digits. But the same mantissas may be found in the 
larger table. 

TaBLE XXV.—The logarithmic sine, tangent, 
etc., of an arc is the logarithm of the natural sine, tangent, 
etc., of the same arc, but with 10 added to the character- 
istic to avoid negatives. This table gives log sines, tangents, 
cosines, and cotangents for every minute of the quadrant. 
With the number of degrees at the left side of the page are 


EXPLANATION OF TABLES ~ ols 


to be read the minutes in the left-hand column; with the. 
degrees on the right-hand side are to be read the minutes 
in the right-hand column. When the degrees appear at 
the top of the page the top headings must be observed, 
when at the bottom those at the bottom. Since the values 
found for ares in the first quadrant are duplicated in the 
second, the degrees are given from 0° to 180°. The differ- 
ences in the logarithms due to a change of 1” in the are are 
given in adjoining columns. 

To find the log sin, cos, tan, or cot of a given 
are. Take out from the proper column of the table the 
logarithm corresponding to the given number of degrees 

and minutes. If there be any seconds multiply them by 
' the adjoining tabular difference, and apply their product 
as a correction to the logarithm already taken out. The 
correction is to be added if the logarithms of the table are 
increasing with the angle, or subtracted if they are decreasing 
as the angle increases. In the first quadrant the log sines 
and tangents increase, and the log cosines and cotangents 
decrease as the angle increases. 

Ezxample.—F¥ind the log sin of 9° 28’ 20’. 

Log sin of 9° 28’ is 9.216097 
Add correction 20 < 12.62 252 


Ans. 9.216349 


Example.—Find the log cot of 9° 28’ 20’. 
Log cotan of 9° 28’ is 10.777948 
Subtract correction 20 < 12.97 259 


Ans. 10.777689 


To find the angle or are corresponding to a 
given logarithmic sine, tangent, cosine, or co- 
tangent. If the given logarithm is found in the proper 
column take out the degrees and minutes directly; if not, 
find the two consecutive logarithms between which the 
given logarithm would fall, and adopt that one which corre- 
sponds to the least number of minutes; which minutes take 
out with the degrees, and divide the difference between this 
logarithm and the given one by the adjoining tabular dif- 
ference for a quotient, which will be the required number of 
seconds, 


314 FIELD ENGINEERING 


With logarithms to six places of decimals the quotient is 
not reliable beyond the tenth of a second. 
Example.—9.383731 is the log tan of what angle? 


Next less 9.383682 gives « 13° 36’ 
Diff. 49.00 + 9.20 = 05/3 
Ams. 13° 30° 05703 


Example.—9.249348 is the log cos of what angle? 
Next greater 583 gives 79° 46’ 


Diff. 235 + 11.67 = 20'7.1 
Ans, 79° 46’ 20".1 


The above rules do not apply to the first two pages of 
this table (except for the column headed cosine at top) 
because here the differences vary so rapidly that interpola- 
tion made by them in the usual way will not give exact 
results. 

On the first two pages, the first column contains the number 
of seconds for every minute from 1’ to 2°; the minutes are 
given in the second, the log sin in the third, and in the fourth 
are the last three figures of a logarithm which is the differ- 
ence between the log sin and the logarithm of the number 
of seconds in the first column. The first three figures and 
the characteristic of this logarithm are placed, once for all, 
at the head of the column. 

To find the log sin of an arc less than 2° given 
to seconds. Reduce the given are to seconds, and take 
the logarithm of the number of seconds from the table of 
logarithms, and add to this the logarithm from the fourth 
column opposite the same number of seconds. The sum 
is the log sin required. 

The logarithm in the fourth column may need a slight 
interpolation of the last figure, to make it correspond closely 
to the given number of seconds. 

Kxample.—Find the log sin of 1° 39’ 14/.4, 


1° 39! 147 4 =" 5954'"4 log 3.774838 
add (q — l) 4.685515 


Ans. log sin 8.460353 


EXPLANATION OF TABLES 315 


Log tangents of small arcs are found in the same way, 
only taking the last four figures of (¢ — 1) from the fifth 
column. 

Example.—Find the log tan of 0° 52’ 35’. 

52’ 35” = (3120 + 35”) = 3155” log 3.498999 
add (q — 1) 4.685609 


Ans. log tan 8.184608 


To find the log cotangent of an angle less than 
2° given to seconds. Take from the column headed 
(q+) the logarithm corresponding to the given angle, 
- interpolating for the last figure if necessary, and from this 
subtract the logarithm of the number of seconds in the given 
angle. 

Example.—Find the log cotan of 1° 44’ 22’.5. 

g.+.b 15.314292 
6240” + 22’.5 = 6262.5 log 3.796748 


Ans. 11.517544 


These two pages may be used in the same way when the 
given angle lies between 88° and 92°, or between 178° and 
~ 180°; but if the number of degrees be found at the bottom 
of the page, the title of each column will be found there also; 
and if the number of degrees be found on the right-hand 
side of the page, the number of minutes must be found in 
the right-hand column, and since here the minutes increase 
upward, the number of seconds on the same line in the 
first column must be diminished by the odd seconds in the 
given angle to obtain the number whose logarithm is to be 
used with (g + /) taken from the table. 

Example.—Find the log cos of 88° 41’ 12’.5. 

(¢ — 1) 4.685537 
4740” — 12.5 = 4727.5 log 3.674631 


Ans. 8.360168 


Example.—Find the log tan of 90° 30’ 50”. 
: g+1 15.314413 
1800” + 50” = 1850’ log 3.267172 


Ans. 12.047241 


316 FIELD ENGINEERING 


To find the arc corresponding to a given log sin, 
cos, tan, or cotan which falls within the limits of 
the first two pages of Table XXV._ Find in the 
proper column two consecutive logarithms between which 
the given logarithm falls. If the title of the given funetion 
is found at the top of that column read the degrees from the 
top of the page; if at the bottom read from the bottom. 

Find the value of (¢—J) or (¢+1), as the case may 
require, corresponding to the given log (interpolating for 
the last figure if necessary). Then if g = given log and 
1 = log of number of seconds, 7, in the required arc, we have 
at once 1 =q—(q—l) or l= (q+l) —q, whence n is 
easily found. 

Find in the first column two consecutive quantities between 
which the number n falls, and if the degrees are read from 
the left-hand side of the page, adopt the less, take out the 
minutes from the second column, and take for the seconds 
the difference between the quantity adopted and the number 
nm. But if the degrees are read from the right-hand side of 
the page, adopt the greater quantity, take out the minutes 
on the same line from’ the right-hand column, and for the 
seconds take the difference between the number adopted 
and the number n. 

Example.—11.734268 is the log cot of what arc? 





pate 15.314376 
q | 11.734268 

Q = 3802’'.8 3.580108 

Forl° adopt 38780. giving 03’ 

Difference 22/’.8 


Ans. 1° 03’ 22”. 8-or 178° 56’ 37’".2, 
Example.—8.201795 is the log cos of what arc? 


pet 4.685556 
: . 8.201795 

nN = 3282.8 3.516239 

For 89° adopt 3300. giving 05’ 

Difference 47/42 


Ans. 89° 05’ 17.2 or 90° 54’ 4277.8, 


EXPLANATION OF TABLES oly 


TaBLE XXVI.—Contains logarithmic versed sines and 
external secants for every minute of the quadrant, with 
the differences of the same corresponding to a. change of ‘1 
second in the arc or angle. Interpolation for seconds is 
made in the same manner as with log sines of the pre- 
ceding table, except on the first two pages. For angles less 
than 4° the differences vary so rapidly that interpolation 
by direct proportion will not give exact values. 

On the first two pages the column headed q — 2/ contains 
the difference between the log versed sine (or log ex secant) 
of an are and twice the logarithm of the number of seconds 
in the same are. The characteristic, and first three decimals 
(9.070) are common to all the logarithms in these columns 
up to 3° 19’ for log vers sines, where it changes to (9.069), 
as shown at the foot of the column; and up to 4° for log ex 
secants, where it changes to (9.071). At the point of change 
a cipher is replaced by the mark * to call attention. 

To find the log vers sin, or log ex sec of an 
angle given to seconds. Reduce the angle to seconds, 
take the logarithm of this number, multiply it by 2, and add 
the product to the logarithm in the column (q — 2l) found 
opposite the given angle. The log (g — 2l) should be 
corrected by interpolation for the fractional part of a minute 
in the given angle. 

Example.—What is the log ex secant of 2° 14’ 43’’.7? 


2° 14’ 43.7 = 8040” + 438.7 = 8083’”.7 log 3.907610 
9 


21 7.815220 
(¢ — 21) 9.070398 


Ans. -. q 6.885618 


To find the arc corresponding to a given log 
vers, or log ex sec. Find in the column of log vers, or 
log ex sec the two values between which the given log falls, 
and take out from the column (q — 2l) the logarithm corre- 
sponding to the given log (interpolating for the value of the 
last figure if necessary). Subtract this from the given 
logarithm and divide by 2. The quotient is the logarithm 
of the number of seconds in the required arc. 


318 FIELD ENGINEERING 


Example.—7.344728 is the log vers of what arc? 








q 7.344728 
3° 48! + (q — 21) 9.069960 
2)8 274768 
13720’.9 “. 1 4.137384 
13680. 
Ans. 3° 48’ 40’.9 


To find the log ex sec of an are greater than 
88° given to seconds. Take from the column (q +l) 
the logarithm corresponding to the given arc, interpolating 
for the fraction of a minute. From this subtracé the log- 
arithm of the number of seconds in the complement of the 
given are. 

Example.—What is the log ex see of 88° 24’ 20’’.5? 


For 88° 24’ q+l 15.302183 
Correction 129 x ee = 44 


q +1 15.302227 
Comp. 88° 24’ 20.5 = 5739/5 log 3.758874 


Ans. log ex see 11.543353 


To find the angle corresponding to a given 
log ex sec when the angle is greater than 88°. 
Find in the table the two consecutive log ex secants between 
which the given one falls, and then find by interpolation 
the value of the log (¢ + J) corresponding to the given log 
ex sec and subtract the latter from it. The difference will 
be the logarithm of the number of seconds in the complement 
of the required angle, which is then easily found. 

Example.—11.924368 is the log ex sec of what arc? 

Given log ex sec 11.924368 


Next less 11.918290 g +t 15:309225 
Diff. 6078 
Correction = prea e P200, xX 6078 = 71 


29141 — 18290 
q +1 15.309296 


Given log ex sec 11.924368 
0° 40’ 26’.2 = 2426/.2 log 3.384928 


~ Ans. 89° 19’ 33/8, 


EXPLANATION OF TABLES 319 


Taste XXVII.—Contains natural sines and_ cosines, 
to five places of decimals for every minute of the quadrant. 
Corrections for fractions of a minute are made directly 
proportional to the difference of consecutive values in the 
table; positive for sines, negative for cosines. 


Taste XXVIII.—Contains natural tangents and cotan- 
gents to five places of decimals for every minute of the 
quadrant. Corrections for fractions of a minute are made 
directly proportional to the difference of consecutive values 
in the table; positive for tangents, negative for cotangents. 


TABLE X XI X.—Contains natural versed sines and exter- 
nal secants to five places of decimals for every minute of 
the quadrant. Corrections for fractions of a minute are 
made directly proportional to the difference of consecutive 
values. They are positive in every case. 


TaBLE XXX.—Contains the number of cubic yards con- 
tained in prismoids of various side slopes, bases, and depths, 
as indicated by the title and the numbers in the first column. 
Each prismoid is supposed to have a uniform level cross- 
section throughout. These tables are chiefly useful in making 
up preliminary estimates from the profile, or in other cases 
where only approximate results are required. For monthly 
and final estimates more elaborate tables are required, such 
as are described in §§ 276, 277, 278 and 279 of the text. 

To make an approximate estimate of quanti- 
ties from a profile by use of Table XXX.—Select the 
proper column for: base and slopes, and if the outline of a 
cut on the profile is roughly a four-sided figure, stretch a 
fine silk thread over the surface line to average it, note the 
depth from thread to grade line midway of the cutting, 
and multiply the tabular number opposite this depth by the 
average length of the cutting in stations of 100 feet. (By 
average length is meant the half sum of the length of the 
grade line in the cutting and of so much of the surface line 
as is covered by the thread.) If the ares of a cutting as 
seen on the profile is approximately ‘triangular, stretch an 
averaging line over each incline, and note the depth from the 
intersection of these lines to grade, and multiply the tabular 
number opposite this depth by one half the length of the’ 
cut measured on the grade line in stations. The resulting 


320 FIELD ENGINEERING 


quantities will be slightly in excess if the ground is level 
transversely, but may be too small if the transverse slope 
is steep, and cutting on the center line is small. In general 
they furnish a good approximation. Quantities in embank- 
ments may, of course, be found similarly. A cut or fill may 
be divided on the profile into several portions, and the 
contents of each portion found separately if preferred. 

The content of a prismoid, level ‘transversely, but having 
different end-depths, may be found correctly by this table 
thus: add together the quantities opposite each end-depth 
and four times the quantity opposite the half sum of the 
depths; multiply the sum by the length in feet, and divide 
by 600. 


TaBLE XXXI.—The volume of a triangular prism is 
equal to the product of the area of the base and the altitude. 
This, in cubic*yards, for a length of 50 feet, is expressed by 


the formula, S = 5 ba, where b and a are the base and 


altitude of the triangular section. Any end area of a section 
of earthwork, whether three-level or irregular, may be 
divided into a number of triangles, and the total volume 
of the solid between the sections may be considered as com- 
posed of prisms with these triangles as bases. The mean 
area volume of a solid for 100 feet in length will be the sum 
of the two 50-foot volumes. An example will illustrate the 
use of the table. 
Example.—Given the cae notes. 


16.0 118 
19.0 _13.6_ 
» 
Btaiw2 70 + 6.0 + 3.0 Bey 


Base 20 feet, slope 13 to 1. 

Two methods might be used for the work. First, where 
the section is divided into four triangles, and second, where 
the area is composed of two triangles less a third triangle 
- which is formed by prolonging the side slope lines to their 
intersection. The former method will be used. 

The cone for station 27 for boar 50 feet is 


27. 0)(19.0 + 13.6) +? 2716.0 + 2.4)(10) 


EXPLANATION OF TABLES 321 


Opposite H’ g’t. 3.0 under 3 take 10 X 8.333 = 83.330 
oS Laldisrsntty lo Xt 46.556 5.556 
“6 SS OS K 16.667 = 1.667 


II 


Also 

Opposite H’g’t. 84 £6 1 ‘* 10 X 7.778 = 77.780 
“3 the total volume for length 50 feet = 168.333 
BB the volume for station 28 for length 50 feet is 


a2. pa + 11.8) +3 27 (4.0 + 1.2)(10) = 109. Bai 


The work may be evlean shortened by not strictly 
observing the terms ‘height’ and ‘‘ width,” since 


54 (32-8) (3.0) is the same in value as the expression given 


above for station 27. This is obtained by entering the table 
once, considering 32.6 as the height, and 3.0 as the width. 

The mean area volume will then be 

168.333 + 109.927 = 278.26, or 278.3 cubic yards. 

The prismoidal correction may be applied to this result, 
thus giving the prismoidal volume. The explanation of 
the method of finding this correction is given below. 

TaBLeE XXXII.—This table gives the correction to be 
‘applied to the mean area volume to give the prismoidal 
volume. The table is based on the formula 


G = 55y d - dD - D’) 
where d and d’ are the center heights, and D and D’ the 
distances between slope stakes at two adjacent stations. 
Using the same data as given under the explanation of 
Table XX XI, the value of the correction is 
= Pm 2G. 4 
GC 554 (8.0 2.4) (32.6 a ag 
From Table XX XI find 
opposite D — D’, 4.8 urider 6 take 4, X 8.889 = 0.9 
The mean area volume from the same example was 278.3 
cubic yards. The prismoidal volume will then be 278.3 — 
0.9 = 277.4. 
The volume for any other distance between sections may 
be found by taking the proper percentage of the volume 


554 a (0.6) (4.8) 


Soe FIELD ENGINEERING 


for 100 feet distance, the multiplication being made after 
the application of the prismoidal correction. 


TABLE XXXIII.—Contains the recommendations of the 
American Railway Engineering Association on minimum 
lengths of spirals for various speeds and degrees of curve. 


Taste XXXIV.—This table exhibits the features of the 
original spiral from which others may be derived. The 
degree of curve is assumed to vary with each chord, and from 
this the central spiral angle up te any point is easily derived, 
and also the inclination of any chord to the main tangent 
drawn through S as meridian. 

The coordinates, x and y of any spiral point, referred to 
S as an origin, are obtained by summation of departures 
and latitudes of the intervening chords. These are given 
with great precision, so that their continuous summation 
may not lead to any error in the final result. 

Finally, the deflection angle from the tangent at S to any 
spiral point is found by dividing x by y for that point, 
giving the tangent for the deflection. All these results are 
tabulated, as also the radius corresponding to the degree 
of curve, the chord length being 100 feet, as with an ordinary 
circular curve. These deflections remain the same for any 
spiral whatever. The variation is made in the length of the 
chord, and the values of x, y, and Rs vary directly with the 
chord length. The value of Ds is derived from Rs. This 
table is made the basis of the tables which follow. 

TABLE XXXV.—This gives the list of deflections to each 
of the spiral points, not only from S (as in the preceding table), 
but also from each one of the twenty other spiral points 
in the system. 

TABLE XX XVI.—This gives a series of spirals of assumed 
chord length from 10 to 50, with the corresponding values 
of Ds, y and x. In any case only so many chords will be used 
as will lead up to (and not beyond) the selected degree of 
curve which is to follow the spiral. The terminal point 
L of any spiral indicates the values of x and y to be used 
in connection with it. 

TaBLE XXXVII.—This table gives certain trigonometric 
functions of the spiral angle s for convenient reference in 
using spiral formulas. 


EXPLANATION OF TABLES 323 


TABLE XX XVIII.—This table, as a companion to XXXVI, 
gives the spiral tangents and long chord for the same set of 
spirals, and also the coordinates, p and q, from S.. Read the 
note at the head of the table. 


TaBLE XXXIX.—Gives a selection of circular curves, 
varying in. degree from 1° 40’ to 35° and suitable spirals 
to be used with each, and also the coordinates p and qg from 
S of the point A’ where the particular curve would meet 
a parallel tangent. Generally there is considerable choice in 
the length of spiral. 


TaBLeE XL.—Exhibits a series of special cases where it 
~ is required to replace an existing simple curve by another 
with spirals so chosen that the length of the track shall not 
be sensibly altered. The middle offset, h, is given, also z, 
and the distance d = AS, from old P.C. to P.S., and the new 
degree of curve, D’. 

The series embraces curves of 2°, 4°, 8°, and 16°, and 
central angles from 10° to 80° inclusive, and will be found 
very useful in solving problems of this sort. 


TaBLE XLI.—Contains concise statements of such geo- 
metrical truths as are applicable to the various discussions 
- in this volume. References are given to Davies’ Geometry, 
in which the demonstrations of the propositions may be 
found. . 


Taste XLII.—Contains all the formulas necessary to 
the solution of any plane triangle; also, a select list of mis- 
cellaneous formulas. A few formulas with respect to the 
versed sine and external secant are new. 


TaBLe XLIII.—Contains a list of the more important 
formulas derived in the text. A large number of special 
formulas on simple curves are given, for which no derivations 
are included. The notation is the same as that used in the 
text. 


Taste XLIV.—Contains a variety of useful numbers 
and formulas. The logarithms are here given to seven 
places of decimals. 























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